algebraic aspects of darboux transformations, quantum integrable systems and supersymmetric quantum

563

Algebraic Aspectsof Darboux Transformations,

Quantum Integrable Systems andSupersymmetric Quantum Mechanics

Jairo Charris Seminar 2010Universidad Sergio Arboleda

Santa Marta, Colombia

Primitivo B. Acosta-HumánezFederico Finkel

Niky KamranPeter J. Olver

Editors

American Mathematical SocietyInstituto de Matemáticas y sus Aplicaciones

American Mathematical Society







Editors

563







Editors

American Mathematical SocietyInstituto de Matemáticas y sus Aplicaciones

American Mathematical SocietyProvidence, Rhode Island

Editorial Board of Contemporary Mathematics

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

Instituto de Matematicas y sus Aplicaciones

Primitivo B. Acosta-Humanez, Director

2010 Mathematics Subject Classification. Primary 12H05, 33E30, 81Q60, 81Q80, 82B23,33E99.

Library of Congress Cataloging-in-Publication Data

Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetricquantum mechanics / Primitivo B. Acosta-Humanez...[et al.], editors.

p. cm. — (Contemporary mathematics ; v. 563)Includes bibliographical references.ISBN 978-0-8218-7584-1 (alk. paper)1. Differential algebra.. 2. Darboux transformations. 3. Quantum theory–Mathematics.

I. Acosta-Humanez, Primitivo B.,

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Contents

Preface vii

Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentialsY. V. Brezhnev 1

Darboux transformation, exceptional orthogonal polynomials and informationtheoretic measures of uncertainty

D. Dutta and P. Roy 33

On orthogonal polynomials spanning a non-standard flagD. Gomez-Ullate, N. Kamran and R. Milson 51

On the supersymmetric spectra of two planar integrable quantum systemsM. A. Gonzalez Leon, J. Mateos Guilarte, M.J. Senosiain,

and M. de la Torre Mayado 73

Solvable rational extension of translationally shape invariant potentialsY. Grandati and A. Berard 115

The pentagram map: geometry, algebra, integrabilityV. Ovsienko 127

Jet bundles, symmetries, Darboux transformsE. G. Reyes 137

Explicit higher-dimensional Darboux transformations for the time-dependentSchrodinger equation

A. Schulze-Halberg 165

Elliptic beta integrals and solvable models of statistical mechanicsV. P. Spiridonov 181

v

Preface

Jairo Charris (1939–2003) was a celebrated Colombian mathematician whoworked in the field of orthogonal polynomials and special functions. Every year, asa recognition of his legacy in Colombia and the relevant contributions he made tothis field, an international high-level meeting in mathematics — the Jairo Charrisseminar — is organized in his country. The topics covered in these meetings in-clude (but are not restricted to) those in which Jairo Charris worked during his life.

This volume corresponds to the Jairo Charris Seminar 2010 entitled “AlgebraicAspects of Darboux Transformations, Quantum Integrable Systems and Supersym-metric Quantum Mechanics” which was held at the Universidad Sergio Arboleda, inSanta Marta, Colombia in August 2010. The aim of this conference was to discussrecent developments and several approaches to the algebraic aspects of Darbouxtransformations, quantum integrable systems and supersymmetric quantum me-chanics.

Some papers in this volume are based on the talks delivered by the authors inthe meeting, while the rest of the contributions are invited papers.

The contribution by Brezhnev treats the classical spectral problemΨ′′ − u(x)Ψ = λΨ and its finite-gap potentials as exactly solvable in quadraturesby the Picard–Vessiot approach, without involving special functions. He showsthat the duality between spectral and quadrature approaches is realized throughthe Weierstrass permutation theorem for a logarithmic Abelian integral. From thisstandpoint, he inspects known facts and obtain new ones: an important formulafor the Ψ-function and the Θ-function extensions of Picard–Vessiot fields. In par-ticular, extensions by Jacobi’s θ-functions lead to the (quadrature) algebraicallyintegrable equations for the θ-functions themselves.

In their contribution, Dutta and Roy study information theoretic measures ofuncertainty for Darboux transformed partner potentials of linear and radial har-monic type. In particular, they evaluate Shannon, Renyi and Fisher lengths forthe isospectral partner potentials whose solutions are given in terms of exceptionalorthogonal polynomials, and the results are compared with the corresponding onesfor the harmonic oscillator.

Gomez-Ullate, Kamran and Milson survey some recent developments in the the-ory of orthogonal polynomials defined by differential equations. The key finding isthat there exist orthogonal polynomials defined by 2nd order differential equations

vii

viii PREFACE

that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomi-als. Unlike the classical families, these new examples, called exceptional orthogonalpolynomials, feature non-standard polynomial flags; the lowest degree polynomialhas degree m > 0. In this contribution the authors review the classification ofcodimension m = 1 exceptional polynomials, and give a novel, compact proof ofthe fundamental classification theorem for codimension 1 polynomial flags. Theyalso describe the mechanism or rational factorizations of 2nd order operators as theanalogue of the Darboux transformation in this context. The authors include anexample of a higher codimension generalization of the classical Jacobi polynomials,and perform the complete analysis of the values of the parameter for which thesefamilies have non-singular weights.

The contribution by Gonzalez Leon, de la Torre Mayado, Mateos Guilarte andSenosiain deals with the analysis of two planar supersymmetric quantum mechanicalsystems built around the quantum integrable Kepler/Coulomb and Euler/Coulombproblems. The supersymmetric spectra of both systems are unveiled, profiting fromsymmetry operators not related to the invariance with respect to rotations. It isshown analytically how the first problem arises at the limit of zero distance betweenthe centers of the second problem. It appears that the supersymmetric modifiedEuler/Coulomb problem is a quasi-isospectral deformation of the supersymmetricKepler/Coulomb problem.

Combining recent results on rational solutions of the Riccati–Schrodinger equa-tions for shape invariant potentials to the scheme developed by Tkachuk, Fellowsand Smith in the case of the one-dimensional harmonic oscillator, Grandati andBerard show in their contribution that it is possible to generate an infinite set ofsolvable rational extensions for every translationally shape-invariant potential ofsecond category.

In his contribution, Ovsienko discusses recent results and open problems relatedto a very special discrete dynamical system called the pentagram map. The penta-gram map acts on the moduli space Cn of projective equivalence classes of n-gonsin the projective plane. Its continuous limit is the famous Boussinesq equation.The most remarkable property of the pentagram map is its complete integrabil-ity recently proved for the (larger) space of twisted n-gons. Integrability of thepentagram map on Cn is still an open problem. He discusses the relation of thepentagram map to the space of 2-frieze patterns generalizing that of the classicalCoxeter–Conway frieze patterns. This space has a structure of cluster manifold,and also possesses a natural symplectic form.

The contribution of Reyes deals with some aspects of the geometric approach todifferential equations due to preeminent mathematicians such as Sophus Lie, Gas-ton Darboux and Elie Cartan. He considers some basic aspects of the formulationof differential equations using jet bundles and, as a non-trivial example, he statesGromov’s h-principle and applies it to systems of differential equations. Followingsome recent work on holonomic approximations due to Eliashberg and Mishachev,the author presents a geometric theorem on local existence of approximate solu-tions to PDEs. He then reviews the theory of symmetries of differential equations,

PREFACE ix

with particular emphasis on internal and nonlocal symmetries. He advances a verynatural approach to nonlocal symmetries using exterior differential systems, andhe argues, motivated by calculations carried out in the context of the Camassa–Holm equation, that nonlocal symmetries can be considered as generalizations ofthe internal symmetries introduced by E. Cartan. Finally he explains, using theassociated Camassa–Holm equation, how to derive Darboux transformations usingnonlocal symmetries and pseudo-potentials.

The contribution by Schulze-Halberg is devoted to the construction of Darbouxtransformations for the time-dependent Schrodinger equations in arbitrary spatialdimensions. The Darboux operator that connects a pair of Schrodinger equationsand the corresponding potential difference are obtained in explicit form. An ex-ample in (5+1) dimensions is presented and the representation of the Darbouxoperator in different coodinate systems is discussed.

The univariate elliptic beta integral was discovered by Spiridonov in 2000. Re-cently Bazhanov and Sergeev have interpreted it as a star-triangle relation (STR).This important observation is discussed in more detail in connection to Spiridonov’sprevious work on the elliptic modular double and supersymmetric dualities. In hiscontribution, Spiridonov describes also a new Faddeev–Volkov type solution of STR,connections with the star-star relation, and higher-dimensional analogues of suchrelations. In this picture, Seiberg dualities are described by symmetries of the ellip-tic hypergeometric integrals (interpreted as superconformal indices) which, in turn,represent STR and Kramers–Wannier type duality transformations for elementarypartition functions in solvable models of statistical mechanics.

Finally the editors would like to acknowledge the assistance of all people in-volved with the Jairo Charris Seminar 2010: Scientific Committee, OrganizingCommitte, Speakers, and also the Instituto de Matematicas y sus Aplicacionesand the Universidad Sergio Arboleda at Santa Marta as host institutions for thisJairo Charris Seminar.

Primitivo Acosta-Humanez, Universidad del Norte, Barranquilla – COLOMBIAFederico Finkel, Universidad Complutense de Madrid, Madrid – SPAIN

Niky Kamran, McGill University, Montreal – CANADAPeter J. Olver, University of Minnesota, Minneapolis – USA

Editors

Contemporary MathematicsVolume 563, 2012http://dx.doi.org/10.1090/conm/563/11162

Spectral/quadrature duality:Picard–Vessiot theory and finite-gap potentials

Yurii V. Brezhnev

Abstract. In the framework of differential Galois theory we treat the classicalspectral problem Ψ′′−u(x)Ψ = λΨ and its finite-gap potentials as exactly solv-able in quadratures by Picard–Vessiot without involving special functions (the

ideology goes back to the 1919 works by J. Drach). We show that duality be-tween spectral and quadrature approaches is realized through the Weierstrasspermutation theorem for a logarithmic Abelian integral. From this standpointwe inspect known facts and obtain new ones: an important formula for theΨ-function and Θ-function extensions of Picard–Vessiot fields. In particular,extensions by Jacobi’s θ-functions lead to the (quadrature) algebraically inte-grable equations for the θ-functions themselves.

Contents

1. Introduction2. Background3. Integrability of equation (1) by Picard–Vessiot4. Spectral/quadrature duality. An integration procedure5. The Θ-functons6. Integration as a linearly exponential Θ-extension7. Integrability and differential closedness8. Definition of θ through Liouvillian extension9. Non-finite-gap integrable counterexamples10. Concluding remarksReferences

1. Introduction

Initially the method of finite-gap integration was developed in works [47, 28,29, 38, 43, 21, 22] as a periodic generalization of the celebrated inverse scatteringtransform method (the soliton theory). In the very first papers on this topic [47,29, 40, 43, 21] it has become clear that analogs and generalizations of the soliton

2000 Mathematics Subject Classification. Primary: 12H05; Secondary: 81Q60.Key words and phrases. Schrodinger equation, finite-gap potentials, Picard–Vessiot theory,

quadratures, Liouvillian extensions, Abelian integrals, theta-functions.Research supported by the Federal Targeted Program under contract 02.740.11.0238 and

partially by Royal Society/NATO and RFBR grant (00–01–00782).

c©2012 American Mathematical Society

1

2 YU. BREZHNEV

potentials are to be smooth and real functions u(x) in the spectral problem definedby the Schrodinger equation

(1) Ψxx − u(x)Ψ = λΨ ,

if the continuous spectrum of the problem consists of finitely many ‘forbidden gaps’1

(lacunae, bands, zones, intervals [43, 38]). This explains the widely used termi-nology ‘finite-gap’, abbreviated further as FG. Ensuing development of the the-ory went far beyond equation (1) and took the algebro-geometric characterization[24, 36, 37, 23]. Riemann surfaces and their theta-functions have become themain subject of study [40, 23, 24]. Appearance of these nontrivial objects is dic-tated by the very nontrivial dependence of solution to Eq. (1) upon parameter λand the search for this dependence is a starting and prime subject of the spectral[43, 28, 29] and algebro-geometric (Θ-function) [36, 23] approaches. These the-ories are referred frequently to as the Θ-function integration. Presently, one cansay that the intense study of equation (1) over the last decades led to the factthat its FG-theory has been developed almost exhaustively. In this connection, itis, perhaps, not without interest to consider one more view on integration of theproblem (1).

1.1. Motivation. In the early 1980s some authors revealed the two old pa-pers by J. Drach [19, 20] wherein equation (1) was integrated ‘directly’ and mainresults of the theory were presented in extremely condensed form. Although theseworks had subsequently received some mention in the literature [25, 16, 45] witha special emphasis to the FG-theory ([8, pp. 84–85]; written by Matveev), somesurprising facet is the fact that more detailed exposition of Drach’s ideology hasnot been presented in the modern literature hitherto. The need for such expositionis apparent when taken into account that the works2 [19, 20] themselves containno any explanations or proofs. In this connection, it is pertinent to make up forthis gap and sketch an appropriate theory.

The original approach by Drach is to integrate (1) as an ODE. Indeed, equation(1) is primarily a differential equation in variable x even though we consider itin the algebro-geometric [36, 7] or spectral context [29, 43] in which the Ψ isviewed, primarily, as function of λ. Anyway, in so far as the Ψ(x;λ) is a functionof two variables, complete theory must explain this duality and therefore provideconversion between ‘x-’ and ‘λ-formulae’. On the other hand, integration of linearODEs is the subject matter of the old and well developed differential extension ofthe algebraic Galois theory which is variously known as the Picard–Vessiot theoryand sometimes as the Lie–Kolchin theory. The main references in this topic aremonographs [49, 30, 9] and classical works [50, 31]. Strange though it may seem,the explicit discussion of a direct linkage between this theory and the modernaspects of integrable models associated to Eq. (1) appeared comparatively recently[44, 2].

Correlation between the Picard–Vessiot theory and Θ-function methods bringsup the following question: what is the relation between these two techniques uponapplying them to the linear spectral problems, say, (1)? The formal answer (com-monly accepted (?)) might be the following: the Θ-series (see next section for

1If parameter λ belongs to such a lacunae (it is a line segment on the real axe λ), then theΨ-function growths unboundedly as a function of x.

2After these works Drach had not longer returned to integration of linear ODEs.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 3

definition) is a special function solving (1). Indeed, an integration theory by Picard–Vessiot begins with the precise definition of a used function class [49, 31]. By thisis meant that, in the strict sense, without definition of the integrability domain, anyscheme based on use of the formally postulated Θ-series should indeed be consid-ered as integration in terms of special functions. We shall show that this is not thecase. For example, the Θ-function integration of Hamiltonian finite-dimensionalnonlinear dynamical systems q = V (q) is know to be a manifestation of their al-gebraically invariant Liouvillian integrability. On the other hand, integrability ofsuch systems is very well known to be related [18] to their representability throughcertain linear equations (Lax pairs):{

L(q)Ψ = λΨ , Ψ = A(q)Ψ}

⇒ L =[A,L

]⇔ q = V (q) .

The ‘nonlinear Hamiltonian’ integrability is certain to entail the ‘linear Picard–Vessiot’ one, if only because the logarithmic derivative Ψ/Ψ is a rational function ofdynamical variables. However, we do not touch here on such Hamiltonian systemsand algebro-geometric (FG) integration of partial differential equations (PDEs).We focus only on a linear problem as such, so our main intention with this workis to show that the scheme of FG-integration of the linear spectral equation (1)should be separated into the two parts:

(1) The invariant property of equation (1) to be integrable, i. e. Lie–Kolchin’ssolvability of corresponding differential Galois group [31, 49].

(2) Representation of differential fields and solutions in terms of those func-tions which of inevitably appear in the theory. These are the Θ-series. Insome particular cases the series themselves satisfy the algebraically inte-grable Hamiltonian ODEs.

By the invariance, here and in the subsequent discussion, we shall informallymean an independence of representations by theta-functions. Notwithstanding thefact that representation of solutions requires introducing the highly nontrivial tran-scendental Θ-objects, the integrability mechanism itself is very simple. It coincidesin effect with an elementary solvability in closed form3 and thereby trivializes un-derstanding of the major portion of the FG-theory. In other words, our mainpurpose is to bring the Picard–Vessiot aspects—fields, their extensions, differentialGalois group, quadrature solvability, etc—into the foreground and, subsequently,to get representations for them in the FG-terms—spectral curves, variables γk, Θ’s,etc. The latter objects, to the best of our knowledge, have not received mentionin the contemporary works on the Picard–Vessiot integration of linear ODEs. See,for example, work [48, first sentence in §3(b)], monographs [49, 44, 9], and quitevoluminous references therein. Partially, some fragments of the theory, in a contextof the elliptic Lame potentials u = A℘(x), can be found in book [44] and work[10]. We note also that FG-potentials are the Abelian functions [40] and Abelianextensions of differential fields were already briefly considered by Kolchin himself[32]. There is no escape from the mentioning nice applications of the Picard–Vessiot

3J. Kovacic, in his famous work [34] on p. 4, notes: ‘by a “closed-form” solution we mean,roughly, one that can be written down by a first-year calculus student’. As we shall see, this‘definition’ is completely compatible with the transcendental Θ-function characterization of theFG-integration of Eq. (1). Rephrasing, there is a closed form solution (Theorem 4.2) that can beverified by a direct substitution into (1) followed by use of the first-year student calculus: algebraand differentiation.

4 YU. BREZHNEV

theory of Eq. (1) to the supersymmetric quantum mechanics. They appeared com-paratively recently and this theme is the subject matter of recent works [1, 2].

1.2. Outline of the work. Section 2 contains the background material: asketch of the classical Burchnall–Chaundy theory of commuting operators and itsmodern formulation in the language of theta-functions.

In Sect. 3 we briefly recall the needed facts from the Picard–Vessiot theory,introduce the base differential fields (Novikov’s fields), and motivate their hyperel-liptic extensions. Then we describe a structure of the differential Galois group forFG-potentials.

Section 4 is devoted to the quadrature (Drach) characterization of the FG-integrability and an explanation as to how the known Weierstrass theorem on anAbelian logarithmic integral performs the transition and difference between quad-rature and spectral mode of getting formulae for the Ψ-function.

In Sect. 5 we first recall that the theta-function formulae can be derived fromquadratures ones [12] and then show the necessity to represent the previous baseobjects in terms of theta-functions and, in particular, to introduce an importantobject—the 1-dimensional section of the theta-function argument with a free pa-rameter. Owing to some differential properties of theta-functions the theory ac-quires very effective form in those cases when jacobians are reducible to a productof elliptic curves.

In Sect. 6 we completely pass to the theta-function representations and givean appropriate formulation to the FG-Picard–Vessiot integrability of Eq. (1). Thisprovides a nice analogy with solubilities in the simplest integrability domains likefield C.

Section 7 explains how the theta-function reformulation of Picard–Vessiot in-tegrability transforms into the differential closedness of the theta-functions them-selves. This also gives a new treatment to the spectral parameter and a relationshipof this treatment to the closedness and linearity of some of defining equations. Weexpound results at greater length for the cases when multi-dimensional Θ reducesto the 1-dimensional Jacobian θ’s. By way of illustration we exhibit a simplestg = 2 non-elliptic potential.

Differential properties of θ-functions described in previous section allows us totake these as a starting point for definition of the functions themselves. It turns outthat such a view leads again to a Liouvillian extension but the latter is accompaniedby introducing a meromorphic elliptic integral and brings up some questions aboutdifferential structures of the multi-dimensional Θ. All this is expounded in Sect. 8.

In Sect. 9 we exhibit some counterexamples fitting no to the canonical FG-theta-theory but being certainly integrable a la Picard–Vessiot with the solvableGalois groups. One of good examples is the famous and fundamental Hermitianequation (containing a parameter) very closely related to the theory of Eq. (1).

Section 10 contains some conclusive remarks.

2. Background

2.1. Commuting operators. The standard soliton/FG-solutions of integra-ble equations are known to be defined through the associated linear PDEs for theauxiliary Ψ-function:

(2) L({U}; ∂x)Ψ = λΨ , Ψt = A({U}; ∂x)Ψ ,


where L and A are the ordinary differential (scalar or matrix) operators with coef-ficients {U} being, in general, some functions of variables (x, t): {U} :=

{uk(x, t)

}.

The set of function {U} is usually termed as the potential. This is because thefirst of Eqs. (2) contains no ∂t and thereby may be considered as a spectral prob-

lem defined by the operator expression L. Moreover, the classical (spectral [29])property of the potential {U} to be finite-gap is determined by the spectrum of thiseigenvalue problem and does not depend on t. On the other hand, the classicalBurchnall–Chaundy–Baker (algebro-geometric) formulation [15, 7] uses one morespectral problem instead of second equation in (2):

(3) L({U}; ∂x)Ψ = λΨ , A({U}; ∂x)Ψ = μΨ ,

In both the formulations the nontrivial theory appears if equations (2) or (3) arecompatible; there exists a common solution Ψ and the potential is subjected to

certain conditions. These are the well-known commutativities of operators[L, ∂t−

A]= 0 for (2) and

[L, A

]= 0 for (3).

2.1.1. Why one should pass from (2) to (3)? There is a simple explanation

as to this question. Let the operator A be determined by an hierarchy of someintegrable PDEs

(4) ∂tU = K([U ]) ,

where, as usual in a formal differential calculus [26] and in the sequel, the symbol [U ]

is used to denote the finite set of derivatives {U , Ux, Uxx , . . .}. Such hierarchies havebeen well tabulated in the literature [18, 27]. Regard Eqs. (2) from the viewpointof their explicit integration (in some sense of the word). In a straightforwardstatement this problem is impossible to solve because t is a hidden variable inthe first of Eqs. (2). In fact, this variable may be thought of as an additionalspectral one4 and the t-dependence of the potential {U} is unknown/undetermined.Complexity of the question is not reduced until Eqs. (2) remain partial differentialequations. Indeed, these equations do not have a general solution expressible interms of any known functions. This is because Eqs. (4), integrable as they are, arenot solvable in general. The only way, in order to solve the question, is to considersome particular situations when PDEs (4) admit transformations into some ODEs.Clearly, such a possibility is related to the self-similar reductions of Eqs. (4) andthe most simple case is of course the reduction to the stationary variable z = x−ct.Assuming now the dependence U = U(x− ct), we get, instead of (2),

L(U(z); ∂z)Ψ = λΨ , Ψt = A(U(z); ∂z)Ψ .

The time t disappears in the first of these equations and therefore there existsa solution in form of separability of variables: Ψ = T (t) · ψ(z). Substituting thisansatz into the last equations, we get immediately a separability parameter5 μ andan exponential dependence T = exp(μt). Equations (2) thus acquire form (3):

L({U}; ∂z)ψ = λψ,(A({U}; ∂z) + c∂z

)ψ = μψ,

where U = U(z). Compatibility conditions of these equations[L, A + c∂z

]= 0

generate the stationary Lax–Novikov equations F ([U ]) = 0 [27] and, incidentally,

4This is so indeed because any additional parameter in coefficients of L({U}; ∂x) may beformally viewed as a spectral variable defining a spectral operator pencil.

5An independent treatment of the second eigenvalue of the second operator.

6 YU. BREZHNEV

different kind reductions may lead to other kind equations. For example, the general

form of Painleve equations and their ‘(L, A)-pairs’ can be obtained by this way [46].2.1.2. Algebraic curve and integrals of Lax–Novikov equations. Let n, m be the

orders of the operators L, A respectively. Assuming that Eqs. (3) are compatible,we conclude, by elimination of the Ψ from (3), that parameters λ and μ are related

by a polynomial dependence W (nμ,

mλ) = 0 and commuting operators L, A them-

selves are also tied by the same dependence: W (A, L) is a zero operator [15, 7].

The identity W (A, L) = 0 implies that its coefficients, being the differential func-tions Ej of {U}, must be free constants Cj independent of x. These yields a setof (compatible) ODEs Ej([U ]) = Cj providing some integrals of motion Ej for theLax–Novikov equations mentioned above [18]. The further theory requires that thepotential {U} be the complex analytic function of the complex argument x.

2.2. Theta-function formulae. Solution to Eqs. (3) is an n-valued functionof λ (and an m-valued function of μ) and this multi-valuedness is related to thealgebraic equation W (μ, λ) = 0. This equation, being viewed as an algebraic curveover C, defines a compact Riemann surface R of a finite genus. It is well knownthat multi-dimensional Θ-functions are the universal tool in order to impart a‘single-valued form’ to the analytic apparatus on Riemann surfaces of multi-valuedalgebraic functions [6]. Baker [7], initiated by work [15], was the first to transferthis λ-multi-valuedness of the Ψ into a single-valued function of a point P ∈ Rand to construct the function itself through Riemann’s Θ-functions; the very firstsentence of the work [7] indicates this. Akhiezer [3] arrived at the same objects whenconsidering the problem (1) from completely different—purely spectral—viewpoint.In the 1970s all these discoveries were substantially developed, generalized [28,29, 40, 22, 36, 37, 24], and acquired their current Θ-function form. The recentexcellent survey by Matveev [42] is, perhaps, the best work both on the historyof the question and background material. Most general and modern treatment ofthese results goes back to works by Krichever and reads as follows.

The spectral variable is thought of as the meromorphic function λ = λ(P) inthe sense that the complex number λ in problem (2) is replaced with an abstractcoordinate on R: the point P. The variable x is viewed as a parameter now andthe function Ψ, as function on R, is the function Ψ(P) of an exponential type[3, 7] with essential singularities of some prescribed form [40, 36]. The formularealization of this result is known presently as a concept of the Baker–Akhiezer

(BA) function [8]. The operator L as above and its coefficients (the potential {U})is called the finite-gap or algebro-geometric. Of course, we could equally well say

the same about μ = μ(P) for operator A in problem (3).The structure of solutions is universal [7, 36]. For example, the scalar problem

(5) LΨ :=dn

dxnΨ+ u2(x)

dn−2

dxn−2Ψ+ · · ·+ un(x)Ψ = λΨ ,

and therefore equation (1), in the class of FG-potentials, has a solution which isgiven, under some normalization, by a general formula:

(6) Ψ(x; λ(P)

)=

Θ(xU +D +U(P)

)Θ(xU +D

) eII(P)x .

Distinctions between different FG-potentials {uk(x)} are only in the changes of theassociated algebraic curve W (μ, λ) = 0 and the curve itself (its R with a canonical


homology base (a,b)) determines all the quantities appearing in (6). Namely: Θ(z)is the canonical g-dimensional theta-series

(7) Θ(z) := Θ(z|Π) =∑

N∈Zg

eπi〈ΠN ,N〉+2πi〈N ,z〉

of g arguments z = (z1, . . . , zg), built by an a-periods Π-matrix of normalizedholomorphic Abelian integrals U(P) =

(U1(P), . . . ,Ug(P)

)on R; symbols like

〈N , z〉 denote Euclidian scalar product 〈N , z〉 :=Njzj ; P is a free point on R;II(P) is a normalized Abelian integral of the second kind with only pole of the firstorder at a point P∞ at which λ(P∞) = ∞; the vector U times 2π i is a vector ofb-periods of this integral; D is an arbitrary constant g-vector. All this terminologyis exhaustively expounded in the numerous literature (see, e. g., [6, 8, 17, 23, 24,27, 40]) and the algebraic dependence W (μ, λ) = 0 is referred frequently to asthe spectral curve. As for the problem (1), this curve constitutes a hyperellipticequation of the form

(8) μ2 = (λ− E1) · · · (λ− E2g+1)

and all the FG-potentials are given by the famous Its–Matveev formula [40, 28]

(9) u(x) = −2 d2

dx2lnΘ(xU +D) + const .

Example 1. Most simple and popular example is a 1-gap potential. It is uniqueand is determined by the Weierstrass elliptic curve

(10) μ2 = 4(λ− e)(λ− e′)(λ− e′′);

we denote its modulus as Π = ω′/ω, where ω, ω′ are Weierstrassian half-periodsnormalized by the condition �Π > 0 [4]. Since this case is a 1-dimensional one, wereplace P → u and put6 U(P) = u. Therefore

II(u) = ζ(2ωu)− 2ηu ⇒ II(u+ Π) = II(u) +π

iω⇒ U = − 1

2ω,

where ζ(z) := ζ(z|ω, ω′) and η := η(ω, ω′) are the standard objects accompanyingthe theory of Weierstrassian function ℘(z) := ℘(z|ω, ω′) [4, 54]. Putting for sim-plicity D = 0, formulae (6) and (9) become

u(x) = −2 d2

dx2lnΘ

(x

2ω|Π)− 2

η

ω, Ψ(x;λ) =

Θ( x

2ω− u

∣∣Π)Θ( x

2ω

∣∣Π) e{ζ(2ωu)−2ηu}x

and λ = ℘(2ωu) (this is the formula λ = λ(P) above). This potential is a preciseequivalent of the classical Lame form u(x) = 2℘(x − ω − ω′) and 1-dimensionalΘ(z|Π)-series here coincides exactly with the Jacobi function θ3(z|Π) defined by thestandard formula (31).

All the constructions mentioned above—spectral, algebro-geometric, Θ-function,and their varieties—are completely equivalent [37], which is why we shall refer tothese approaches merely as spectral for short.

6Jacobian of an elliptic curve is isomorphic to the curve itself.

8 YU. BREZHNEV

3. Integrability of equation (1) by Picard–Vessiot

3.1. The function R. The theory of equation (1) is closely related to thefundamental linear differential equation

(11) Rxxx − 4(u+ λ)Rx − 2uxR = 0

determining function R = R(x;λ). Integrating this equation and denoting an inte-gration constant as μ, we get

(12) μ2 = −1

2RRxx +

1

4Rx

2 + (u+ λ)R2

(see work [12] for a successive derivation of these formulae with use of Lie’s sym-metries approach.) Then the FG-solutions to the Ψ-function for Eq. (1), in generalposition μ �= 0, are given by the well-known formula

Ψ±(x; λ) =√

R(x;λ) exp

x∫ ±μdxR(x;λ)

= exp

x∫Rx(x;λ)± 2μ

2R(x;λ)dx.(13)

Formulae (11)–(13) and their variations have been repeatedly appeared in the lit-erature [22, 24, 5, 8, 27]. Their precise meaning, however, lies in the fact that theavailability of formula (13) itself does not mean any integrability [12]. This is justan ansatz for Eq. (1) and its solution should be written down in terms of indefiniteintegrals; otherwise all the procedure would reduce to re-notations.

3.2. General formula for the Ψ-function. In the language of commutingBurchnall–Chaundy operators [15] an explicit formula for the Ψ-function (including(13)) results from the sequential elimination of derivatives Ψ(k) from the pair ofdifferential equations (3) down to the formula

(14) Ψx = G([U ];λ, μ)Ψ ⇒ Ψ = exp

x∫G([U ];λ, μ)dx.

This simple recipe of getting the Ψ-function, perhaps, has no received mention inthe modern literature [53]; it is, however, implicitly exploited in monograph [27] .Elimination of the last derivative Ψx leads to equation of the curve W (μ, λ) = 0.

The algebraic dependence W (A, L) = 0 serves, in some of works, as the basis for aformal definition of the algebraic integrability [37].

3.3. Novikov’s equations and differential field. Motivated by the desireto define a differential field over which the integration is performed, we need to knowthe differential structure of function R(x;λ). It is well known that this function isa series in λ with coefficients being differential polynomials in u(x). Computationalformulae for these polynomials have been detailed in the famous work [26, formulae(8)]. Finite order differential conditions on the potential appear if equation (11)has a solution being a polynomial in λ [12]. As in the FG-theory, the equation(11) is also known in differential Galois theory as the second symmetric power ofoperator (1) [49, §4.3.4], [51, p. 671].

Definition 3.1. The potential u(x) is said to be an FG-potential if equation(11) has a solution R(x;λ) being a polynomial in λ. No restrictions on coefficientsof the curve (12) ⇔ (8) have been imposed.


This definition is not a standard one but it is of course equivalent to the spectral[29, Theorem 1], algebraic [26, Ch. 3], or quadrature [20, 12] definitions. The onlyexception is a formal Θ-function (algebro-geometric) setting because it does not usespectra, resolvent R, or quadratures. An important point here is the fact that thefunction domain for the potential is not defined as usual in Galois theory but calcu-lated. This calculation is a problem of nonlinear integration, that is integration ofODEs for the potential u(x). These are the famous Novikov equations [47]. We shallcall this base field the Novikov differential field N ([u]) in u(x)-representation bear-

ing in mind that u(x) satisfies a Novikov equation F (u, ux, uxx , . . . , ux(2g+1)) = 0.

The field N ([u]) consists of rational functions of u(x) over C(λ) and its derivatives.Parameter λ, the g constants ck coming from the integral Gel’fand–Dickey recur-rence [26], and branch points Ek of the curve (12) belong to a subfield of constantsfor N ([u]). Here is an example of Novikov’s equation under g = 2:

(15) (uxxxxx − 10uuxxx − 20uxuxx + 30u2ux) + c1(uxxx − 6uux) + c2ux = 0 .

Integration constant μ, in an FG-class, is fixed to be dependent on the pa-rameter in equation, that is μ = μ(λ), and equation (12) turns into the formula(8). At the same time, as soon as R(x;λ) becomes a polynomial in λ it becomesa differential polynomial R([u];λ) ∈ N ([u]). It should be noted that one suffices tohave only one solution of Eq. (11) belonging to N ([u]) (see Sect. 9.1 further below).

3.4. Picard–Vessiot field, constants, and their hyperelliptic exten-sion. As usual, the Picard–Vessiot extension

N ([u])〈Ψ±〉 :=N ([u])(Ψ+,Ψ−,Ψx+,Ψx

−) ,

i. e. the splitting field, results from attaching to the field N ([u]) integrals Ψ± ofequation (1) and its derivatives [31, 50, 33, 30, 44]. A simplest kind of extensionscorresponds to solvable cases of the Galois theory and is known as the extension ofLiouville [49, p. 33], [44, 30].

Definition 3.2. An extension N of the differential field N is said to be Liou-villian if there exists a tower of fields N = N0 ⊂ N1 ⊂ · · · ⊂ Nn = N such thatNk+1 = Nk(ψk), where ∂xψk or ∂x lnψk is an algebraic element over Nk.

In other words, Liouvillian extensions are the natural enlargements of the basefield performed by a step-by-step adjunction of solution to the simplest integrableODEs: the 1st order linear ODEs

(16) ψx = Aψ + B

with coefficients (A,B) being algebraic/rational over previous step field. The stan-dard adjunctions of an algebraic element a, integral

∫pdx, or exponential exp

∫pdx,

where p ∈ Nk, are obtained by putting here (A,B) = (0,ax), (A,B) = (0, p), and(A,B) = (p, 0) respectively.

Solvability of equation by quadratures is directly connected with a structure ofthe group constituting a differential version of the polynomial Galois group.

Definition 3.3. Differential Galois group Gal(N〈Ψ〉

)of a linear ODE de-

fined over N is a set of linear transformations of its solutions Ψ’s that preserve allthe algebraic (over N ) relations among Ψ’s and their derivatives Ψ(n) (differentialautomorphisms group).

10 YU. BREZHNEV

The Picard–Vessiot extension must have the same set of constants as N ([u])[31], [33, p. 411], [49]. It is known that for equations of the form (1) the WronskianΨ′

1Ψ2 −Ψ′2Ψ1 is a constant. Taking (13) as one of possible bases for solutions, we

get

(17)Ψx

+Ψ− −Ψ+Ψx− =

√Rx

2 − 2RRxx + 4(u+ λ)R2

= 2μ(λ) .

Clearly, it must be a constant of the Picard–Vessiot field whatever the potentialu(x) may be. Neglecting for the moment other constants, we obtain that field ofconstants C(λ) requires adjunction of the constant μ: C(λ) → C(λ, μ). Characterof this extension is determined by the λ-dependence of the function R. It may bepolynomial, rational, or essentially transcendental. The case of rational polynomialR(λ) is possible only if its poles do not depend on x; for we should otherwise havethe λ-dependent poles of the Ψ-function, which is impossible by virtue of structureof Eq. (1). Hence the function R must be an entire function of λ. The Galoisgroup does depend in general on parameters of equation but we are interested inthe following cases:

• When the Lie–Kolchin integrability structure is the same for generic λ?

Therefore two kinds of theories do exist, according as the field C(λ, μ) does not,or does, belong to a finite algebraic or infinite extension of C(λ). The latter fieldsare excessively general because there are huge varieties of entire transcendentalfunctions (without any classification) and they do not produce any restrictions onpotential. On the other hand, finite extensions are of fixed algebraic (necessarily hy-perelliptic) character and calibrated by the only number, namely, by the λ-degree gof the polynomial R([u];λ). In this case infinite Gel’fand–Dickey recurrences termi-nate and equation (12) leads to a finite set of differential restrictions on u(x) in formof differential polynomials Ej([u]) = Cj . This gives in fact yet another indepen-dent motivation (a la Picard–Vessiot) for appearance/availability of a polynomialsolution to Eq. (11). For brevity, we shall adopt however the previous shorter no-tation for the base field N ([u]) without explicit indication of its λ, μ-dependenceN ([u];λ, μ) or dependencies on other field constants N ([u];λ, μ, Cj, . . .), where dotsstand for remaining constants which arise upon complete integration of a Novikovequation.

3.5. Finite-gap Galois groups. Below is a characterization of the Galoisgroup of equation (1) in the class of FG-potentials. Condition on parameter λ ofbeing an arbitrary quantity is a fundamental requirement meant throughout thepaper.

Theorem 3.4. The Picard–Vessiot extension N ([u])〈Ψ±〉 is a Liouvillian ex-tension of the transcendence degree equal to 1. Associated group Gal

(N ([u])〈Ψ±〉

),

under generic λ �= Ej, is connected and isomorphic to the group G =(α 0

0 α−1

),

where α ∈ C. For other values of λ it is isomorphic to group G =(±1 α

0 ±1

).

Proof. For generic λ’s integral (13) does not belong to N ([u]). From (13)it follows that this extension is Liouvillian. Take the canonical basis of solutions(13). Then the quantities {Ψ−,Ψx

±} are expressed through the transcendent being


adjoined Ψ+ as follows:

(18) Ψ− =R

Ψ+, Ψx

+ =Rx + 2μ

2RΨ+ , Ψx

− =Rx − 2μ

2Ψ+(μ �= 0) .

Clearly, in the case of such λ’s that μ = 0 these relations cease to be valid since(Ψ+, Ψ−) become linearly dependent on each other. The relations should be mod-ified and we choose the following basis Ψ±:

(19) (Ψ−)2 = R, Ψx− =

Rx

2RΨ− , Ψx

+Ψ− −Ψ+Ψx− = 1 .

In both of these cases we adjoin one transcendent element (respectively):

Ψ+ = exp

x∫Rx + 2μ

2Rdx or Ψ+ =

√R

x∫dx

R.

In the latter case the radical√R = Ψ− can sometimes be element of N ([u]) as an

example of the Lame equations shows [54, Ch.23], [39].Let us check invariance of relations (18)–(19) with respect to linear transfor-

mation of the basis (Ψ+

Ψ−

)→

(αΨ+ + βΨ−

γΨ+ + δΨ−

);

this determines the Galois group G =(α βγ δ

)completely. In case (18) we get the

following set of equalities:

αγ = 0 , β δ = 0 , αδ + βγ = 1 , β = 0 .

From this it follows that β = 0, γ = 0, and αδ = 1. The number α can not be anyalgebraic one; for we should otherwise have a finite Galois group and the algebraicΨ±-solutions to give a contradiction with a single-valuedness of the general formula(6). In case (19) we derive that δ2 = 1, γ = 0, αδ = 1, and no conditions on β(except for degenerated cases of curve). Hence in generic case λ �= Ej the group G

is connected. Its possible forms are thus as follows:

G =

(α 00 α−1

)or G =

(ε β0 ε−1

),

where ε = ±1 (compare with cases 3, 4 in Proposition 2.2 of [44]). �To summarize briefly, we conclude that independently of the topological genus

of the curve (8), ‘finite-gap’ groups Gal(N ([u])〈Ψ±〉

)do not depend on parameter

λ and cease to be diagonal and connected only for isolated values of the parameter.

Corollary 3.5. Equations (1) of the FG-class are factorizable over N ([u]):

∂xx − (u+ λ) =

(∂x +

1

2

Rx

R± μ

R

)(∂x −

1

2

Rx

R∓ μ

R

).

Remark 1. We used nowhere any specific form of the polynomial R. Theo-rem 3.4 is easily restated for arbitrary integrable λ-pencils of the 2nd order withthe only condition that R(x;λ) ∈ N ([u]). Recall that the spectral λ-pencil is a gen-eralization of the canonical spectral equation of the form (5) to more complex (e. g.

polynomial) dependencies of the differential expression L on the external parameter

λ, that is L({U}; ∂x, λ) = 0. An example is the well-known spectral λ-pencil of theform

(20) Ψxx −ux

uΨx −

(λ2 − ux

uλ+ uv

)Ψ = 0;

12 YU. BREZHNEV

it arises when integrating the integrable nonlinear Schrodinger equation [27, 45, 8].In Sect. 9 we shall consider other examples of the λ-pencils (see also [12, Sect. 5(a)]).

By virtue of structure of the formula (14) Theorem 3.4 has a direct generaliza-tion.

Theorem 3.6. Let N ([U ]) be a Novikov field associated with the pair of Burch-nall–Chaundy scalar operators (3). Then both of these equations are integrable inLiouvillian extensions of the same transcendence degree. Under generic λ, μ thedifferential Galois groups of these equations are connected and isomorphic to thediagonal groups G = Diag(α, β, . . . , γ) ⊂ GL(C).

Novikov’s equations are known to be Hamiltonian systems integrable by Liou-ville [18]. However such a way of their integration is not necessary since determi-nation of the potential, i. e. construction of the field N ([U ]), is given by formulaefollowing completely from the ‘linear’ Picard–Vessiot theory. It does not requireHamiltonians.

We finish this section with a digression to one remarkable example. It is aMatveev 1-positon potential given by the seemingly elementary formula [41]

(21) u = −2 lnxx{sin(ax+ b)− ax− c

}.

Surprisingly, in spite of its complete fitting into the integration scheme above, itis not amenable to integration by means of any classical algorithm in the Picard–Vessiot theory (Kovacic [34], Singer [50]). Indeed, these algorithms are applicableto the finite algebraic extensions of C(x), whereas this u ∈ C(x, ei(ax+b), a, c).

4. Spectral/quadrature duality. An integration procedure

4.1. Drach–Dubrovin equations. The quadrature Drach approach gives avery simple explanation as to why and where the fundamental polynomial

(22) R([u];λ) =(λ− γ1(x)

)· · ·

(λ− γg(x)

)comes from, what its roots γk are, and why these are precisely the quantities thatcomplete the indefinite quadratures. In what concerns the spectral viewpoint, aremarkable result by Dubrovin [21, 22] is that the quantities γk arise as zeroes ofa Θ-function since they solve the inversion problem of Jacobi [3, 40, 29].

Lemma 4.1 (Drach [20], Dubrovin [21]). Functions γk(x) satisfy the system ofODEs

(23)dγkdx

= 2

√(γk − E1) · · · (γk − E2g+1)∏

j �=k

(γk − γj), j, k = 1, . . . , g

and potential is determined by the trace formula [28]

(24) u = 2

g∑k=1

γk(x)−2g+1∑k=1

Ek .

Proof. One inserts (22) into (12) and takes (8) into account. Collecting theresult in degrees (λ− γk)

n, one requires identity under arbitrary λ. One gets (23)and (24). �


Remark 2. We presented such a way of proof because it is applicable to higherorder operators and even spectral λ-pencils. The reason is that the derivation oftrace formulae is not evident when generalizing. Formulae of such a kind ceaseactually to be the ‘trace formulae’ because they have no longer Gel’fand’s treat-ment as the operator trace analogs. They are also not derivable directly from thedefinition of R-polynomial like (22); illustrative counterexamples can be found inwork [11].

4.2. Weierstrass theorem and new representation for the Ψ-function.Main content of this section was briefly announced in [12] and we shall presenthere the extensive proofs, derivations, and precise correlation between spectral andquadrature approaches.

Theorem 4.2. Let u(x) be an FG-potential corresponding to the arbitrary curve(8). Then solution to equation (1) is given by the quadratures

(25) Ψ±(x;λ) = exp1

2

{ γ1(x)∫w ± μ

z − λ

dz

w+ · · ·+

γg(x)∫w ± μ

z − λ

dz

w

},

where w2 = (z − E1) · · · (z − E2g+1) and functions γk = γk(x) are determinedthrough inversion of the set of indefinite integrals

(26)

g∑k=1

γk∫zg−1 dz

w= 2x+ ag ,

g∑k=1

γk∫zn

dz

w= an+1 , n = 0, 1, . . . , g − 2 .

The FG-potential u(x) is determined by formula (24).

Proof. Let us substitute (22) into (13) and change the integration variable xto z. We then obtain the following rules

Rx

Rdx = d ln

∏k

(λ− γk) �� 1

z − λdz ,

2μ

Rdx =

2μ∏j(λ− γj)

· 12

dγk�k

∏j �=k

(γk − γj) �� −μz − λ

dz

w,

wherein �2k = (γk−E1) · · · (γk−E2g+1). Abelian integrals of 3rd kind, as appeared in(25), result from differential equations (23). Furthermore, substitution of expression(25) into equation (1) leads, to get an identity, to formula (24); this can serve as yetanother way of derivation of the trace formula. Symmetrizing right hand sides ofequations (23), we rewrite them down in a form that admits an application of theindefinite integration operations, that is (26). This set determines γk as functionsof x. �

Corollary 4.3. In u(x)-representation the extension N ([u])〈Ψ±〉 requires theone quadrature (13). If formula (25) is used then extensions N ([u])〈Ψ±〉 consist inan adjunction of a symmetric sum of the logarithmic Abelian integrals.

Remark 3 (Definition). Special attention must be given to the fact thatintegrability of the ‘linear Ψ’ is also algebraic (hyperelliptic) as is nonlinear inte-grability of N ([u]). More precisely, in what follows the term ‘algebraic’ will meanthat ultimate answers contain finitely many indefinite integrals of algebraic func-tions and inversions of the formers.

14 YU. BREZHNEV

As for transition from primary λ-dependence to the x-one and vice versa, theduality between spectral and quadrature representations is nontrivial; it is charac-terized by the following statement.

Theorem 4.4. Quadrature and spectral approaches are equivalent in the sensethat the explicit transition between formula (25) and its spectral counterpart (seebelow) is realized through the Weierstrass theorem on permutation of limits andparameters in a normalized Abelian integral of third kind.

Proof. Sub-exponential expression in (25) is a sum of Abelian integrals, eachwith logarithmic singularities on R at two points: (z, w) = (λ,+μ) and branchplace (z, w) = (∞,∞). According to a Weierstrass theorem, we may exchangesingularities of an elementary logarithmic integral with its limits [6, 17, 40]. Inour case, these are (z, w) = (γ(x), �(x)) and (z, w) = (α,+β), where β2 = (α −E1) · · · (α− E2g+1). More precisely, switching the places{

(λ, μ)(∞,∞)

}�

{(γ(x), �(x)

)(α, β)

},

we obtain that the differenceγ(x)∫α

w + μ

z − λ

dz

w−

λ∫∞

{w + �(x)

z − γ(x)− w + β

z − α

}dz

w= · · ·

is to be everywhere finite quantity, that is certain holomorphic integral:

· · · =γ(x)∫ {

A1(λ) +A2(λ)z + · · ·+Ag(λ)zg−1

} dz

w.

Sum of g such quantities must be a holomorphic integral depending symmetricallyon γ’s. Expression (25) is thus converted to its dual object

(27) Ψ± � exp1

2

{ λ∫w ± �

1(x)

z − γ1(x)

dz

w+ · · ·+

λ∫w ± �g(x)

z − γg(x)

dz

w+holomorhic(λ, x)

}.

This is nothing else but the spectral formula by Its & Matveev [29] deserving tobe mentioned more often. We reproduce7 their result as it has been written in [29,p. 351]:

(28) ω(λ) =

λ∫βn

M(λ)

2√p(λ)

dλ,

αj∫βj−1

dω(λ) = 0, j = 1, . . . , n,

where M(λ) = λn + a1λn−1 + . . .+ an and

ωk(λ) =

λ∫∞

(√P (λ) +

√P (λk(x))

λ− λk(x)−√

P (λ) +√

P (λk(0))

λ− λk(0)+Mk(λ)

)dλ

2√

P (λ),(29)

ψ(x, λ) = exp

(ixω(λ) +

n∑k=1

ωk(λ)

).(30)

7We have not found mention of this important result in the literature. See also formula (3)in Akhiezer’s work [3].


Meaning of all the quantities presented in (28)–(30) and transition (25) � (30)are obvious from the context. To put it differently, the permutation theorem ofWeierstrass, regarding inverse transition (30)→ (25), is a way of doing normaliza-tion of periods of these integrals so that all the parameters in the spectral formulae(28)–(30) can be ‘dumped’ to a common multiplication constant for the Ψ. By thismeans the non-indefinite integrals (27) or (29) with a parametrical dependenceon transcendent functions γk(x) turn into the indefinite ones (25) of an algebraicfunction. �

Description of invariant property of equation (1) to be integrable has beencompleted and we conclude the section with the comments about fundamentaldifference between spectral and quadrature modes of getting the formulae.

4.3. On a Riemann surface. At this point not only do analysis on Riemannsurfaces does not come into play, but also the surfaces themselves do not appear.One has just a designation

μ :=√(λ− E1) · · · (λ− E2g+1)

and the theory consists of elementary substitutions (see footnote on p. 3). On theother hand, verification of spectral formulae (28)–(30) is a highly nontrivial tasksince they contain a complete set of transcendental objects of Riemann’s theory ofAbelian integrals and differentiation of an integrand containing γ’s. In this respect,not using the permutation theorem, the ‘spectral integral’ in (27), that is

λ∫∞

w ± �(x)

z − γ(x)

dz

w,

would be very akin to an integral representation of any special function, say, com-plete elliptic Legendre’s integral

K(x) =

1∫0

dz√(1− z2)(1− x2z2)

.

The latter is not expressible by means of any finite Liouvillian extension over C(x)since K(x2) satisfies a 2nd order irreducible 2F1

(12 ,

12 ; 1

∣∣x2)-hypergeometric equa-

tion [4].

Remark 4 (history). Both the integrabilities are due to Liouville but chrono-logically, ‘linear integrability’ (1830–40s) was preceded by more famous nonlinearintegrability of Hamiltonian systems (1840–50s). Despite the numerous modernliterature, the fact that these two kinds of integrability are non-casually related tothe one name Liouville was first observed by Morales-Ruiz [44, pp. 51–52].

5. The Θ-functons

By virtue of the fact that extension N ([u])〈Ψ±〉 is transcendental, analyticrepresentations for the previous formulae require introducing new functions. Theseare the Θ-series (7) involved to the theory by Matveev and Its in their famous work[29].

Theorem 5.1. The Θ-function representations (6), (9) are deducible fromquadrature (25). The expressions (6) and (25) are proportional to each other.

16 YU. BREZHNEV

Proof of this theorem and consecutive derivation of formula (6) from (25) havebeen detailed in [12, §7]. An important point here is the deductive appearance ofall the aggregates of formula (6) when it is viewed as an axiomatic one: the integralII(P), its periods U , and the Abel map U(P).

From this theorem, we may draw the conclusion that when recognizing the in-tegrability of linear equations the Θ-functions themselves are not necessary. Theyrealize a step to be considered as the next one after emergence of an integral sym-bol

∫in (13) and (26).

5.1. Some differential properties of theta-functions. Theorem 5.1 sug-gests a search for representation of the field N ([u]) by means of Θ-functions. To dothis require some differential properties of Θ-functions and we show further thatthey are available. All the FG-theory tells us that Abelian and BA-functions satisfycertain differential identities. By these identities are meant the fact that Abelianfunctions, as theta-function ratios of linear sections of g-dimensional jacobians ofcurves, have a lot of differential relations between themselves and many of suchrelations have forms of known integrable PDEs [14, 27]. Adding here exponential

functions of the BA-type, we involve into analysis (L, A)-pairs for these PDEs.Moreover, let FG-potential be expressible through the θ-functions of Jacobi. Thennot only do Abelian and BA-functions satisfy certain differential identities but θ-functions themselves also satisfy some ODEs.

Denote by θ[εδ

]the standard θ-series of Jacobi with characteristics (ε, δ) [4]:

(31) θ[εδ

](x|τ ) :=

∞∑k=−∞

eπi(k+ ε

2)2τ+2πi(k+ ε

2)(z+δ2) ,

i. e. θ[11

]= −θ1, θ

[10

]= θ2, θ

[00

]= θ3, θ

[01

]= θ4. Let ϑ := θ(0|τ ) be corresponding

ϑ-constants and θ′1 stands for x-derivative of the series −θ[11

](x|τ ). Period of a

meromorphic elliptic integral is denoted by η = ζ(1|1, τ ).

Theorem 5.2. Jacobian functions θ[εδ

], θ′1 with arbitrary integral characteris-

tics are differentially closed over the field of the (ϑ2, η)-constants and satisfy theautonomous ODEs

(32)

⎧⎪⎪⎪⎨⎪⎪⎪⎩∂θ

[εδ

]∂x

=θ′1θ[11

] θ[εδ]− (−1)[δ2 ]επϑ

[εδ

]2 · θ[ε−1

0

]θ[

0δ−1

]θ[11

]∂θ′1∂x

=θ′1

2

θ[11

] − π2ϑ[00

]2ϑ[01]2 · θ[10

]2

θ[11

] − 4

{η +

π2

12

(ϑ[00

]4 + ϑ[01

]4)} ·θ[11] ,where

[δ2

]signifies an integer part of the number δ/2.

These formulae are consequences of more general differential properties of Ja-cobi’s functions briefly tabulated in [13]. One can see that the similar propertiesare inherent characteristics of the general Θ-functions if the g-dimensional jacobianis isomorphic to a product of elliptic curves. Many examples of such reductions canbe found in [8].

Example 2. Define a Θ-function with characteristics[αβ

]as follows:

(33) Θ[αβ

](z|Π) := i〈α,β〉Θ

(z +

1

2Πα+

1

2β∣∣∣Π) · eπi〈α,z+ 1

4Πα〉 .

Then in the case g = 2 we have the following identity.


Proposition 5.3. The reduction formula for genus g = 2 under Π12 = 12 :

(34)

Θ[αεβδ

]( 12 z−

18ατ

w− 14α

∣∣∣ 14 τ

12

12 κ) =

= eπ2 iα(z+β+ 1

4ατ) · {θ[0ε](z|τ )θ[εδ](w|κ) + i2β+ε ·θ[1ε

](z|τ )θ

[ εδ−1

](w|κ)}.

Differential properties of these Θ, Θ′-functions follow completely from Theorem 5.2.

This example is a rather illustrative one because curves have often symmetriesand if a genus-2 curve has an involutory symmetry differed from the hyperellipticone (λ, μ) → (λ,−μ) then one can show that its Π-matrix is reducible to form (34).Formula (34) is perhaps a simplest case of reduction of the theta-functions to thetwo elliptic tori τ and κ. More complex equivalent of (34) is presented in [8].

Corollary 5.4. Let jacobian of the curve (8) be splittable into a product ofthe elliptic curves and the collective symbols θ, ϑ, η stand for arising Jacobi’stheta-functions and their constants. Then the field C∂(θ;ϑ

2, η, . . .) is a differentialextension of N ([u]) (dots indicate other constants of the field).

Example 3. Non-elliptic 2-gap potential for the reduction case (34):

(35) u = −2 lnxx{θ4(Ux+A|τ )θ2(V x+B|κ)− iθ1(Ux+ A|τ )θ1(V x+B|κ)

},

where {τ,κ, A,B, V } are arbitrary. Since the differential θ-calculus is completelyat hand, we get a particular but nontrivial example of solution to Dubrovin’s ef-fectivization formulae for genus g = 2. Recall that the problem consists [23] indetermination of the ‘winding’ vector U and is described by a system of equationscontaining the undetermined fourth derivatives of the Θ-function [23]. In the ex-ample under consideration the ultimate answer turns out to be quite finite butsomewhat large to display here. Equation for the one sought-for quantity U is analgebraic equation of degree 9 (exercise: derive it).

5.2. Linearly exponential divisor and Θ-representation of N ([u]). Letus use notation of formula (6) and plug into (9) an inessential exponential multiplier:

(36) u(x) = −2 d2

dx2lnΘ(xU +D)ehx + const .

Introduction of this term is motivated by the fact that solutions for the Ψ-function(6) are expressed not only through the Θ-functions but involve an exponentialfactor. We shall call the quantity Θ(xU+D)ehx, with h, U , andD being constantswith respect to ∂, the linearly exponential divisor or Lx-divisor. Let us form a fieldC(Lx). The following proposition gives a weaker property than Corollary 5.4, butit is valid for arbitrary genera.

Proposition 5.5. The field C(Lx

)is ∂-differential and finitely generated.

Proof. Expression (36) satisfies a Novikov equation which has finite order

2g + 1. Hence the derivatives dn

dxnΘ(xU + D)ehx of order n � (2g + 1) + 2 areexpressed rationally through Lx and its lower derivatives. �

The field Θ∂ = C∂

(Θ(xU +D)

)may be considered as a field generated by one

linear divisor (h = 0). It is obviously that N ([u]) ⊂ Θ∂ . Of course, Θ∂ containsnow not only Abelian functions but this extension is well defined since Θ-series

18 YU. BREZHNEV

and b-periods U are computed once u(x) has been given. For this reason, we canredefine equation (1) as one given over Θ∂ and thereby we let

(37) N (Θ) := C∂

(Θ(xU +D)

).

Although N ([u]) � N (Θ), the field N (Θ) is said to be a Θ-representation ofNovikov’s fields. The constant λ, for the moment, may be disregarded.

It should be emphasized that in the FG-integration there appears not merelyan abstract multi-dimensional Θ-series but its specification Lx; the 1-dimensionallinear section of jacobians. Moreover, the most general Θ-function is an objectdefined up to an exponential multiplier (see (33)) so that we may thought of it, andtherefore of the divisor Lx, as a continual generalization of Θ-function with discretecharacteristics. Thus, Liouvillian solutions N ([u])〈Ψ±〉 are expressed through theΘ with a linear dependence of its arguments upon x and the ‘linearly exponential’multiplier8 eII(P)x first explicitly appeared in Akhiezer’s work [3] and subsequentlywas axiomatized by Krichever [36].

Remark 5. In what follows we shall exhibit that the continually parametricobjectLx may be introduced in its own right, in a particular case, through the quad-rature integrable ODEs. Interestingly, the different kind sections of theta-functionarguments can lead to other important equations. One remarkable property of sucha kind appears even in the g = 1 case. Let us consider the function θ(x|τ ) as a func-tion on a simplest (i. e. straight line) section of the 2-dimensional variety {jacobian⊗ moduli space}. Without loss of generality we may impart to this function theform θ1(Aτ+B|τ ). Then this object generates the general Hitchin class of solutionsto the sixth Painleve equation in a form exactly as does the finite-gap formula (9),that is logarithmic derivative of a ratio of entire functions [13].

6. Integration as a linearly exponential Θ-extension

In this section we give a formulation of the Θ-function scheme as integra-tion a la Picard–Vessiot. Let us consider the above Picard–Vessiot extension [49]N ([u]) ⊂ N ([u])〈Ψ±〉 in the representation u(x). This transcendence is Liouvillianand dependence of the Ψ on parameter λ is also transcendental contrary to the‘rationality’ of the field C(λ, μ). Meanwhile, based on Theorems 3.4 and 5.1, we seethat the field has the following structure:

N ([u])〈Ψ±〉 = C∂

(Θ(xU +D +U(P)

)Θ(xU +D)

eII(P)x

).

It immediately follows that if we pass to the Θ-representation (37) then integrationprocedure can be reduced to one operation. Namely, a field, over which an equa-tion has been defined, is supplemented with an element of the same form as onegenerating the field itself:

(38) N ([u]) ⊂ N (Θ) ⊂ C∂(Θ(xU +D)ehx, Θ(xU +D +U(P))eII(P)x).It is significant in this viewpoint that problem of the ‘linear integration’ drops

out along with the problem of building the base field N ([u]) being treated as aproblem of the nonlinear integration. In the Θ-representation the potential is de-termined only by means of operations in the field N (Θ); formula (36).

8This ‘linear exponent’ is a result of the contemporary theory; Baker [7] did not specify anexponential Θ-structure of solutions but just cited to pp. 275, 289 of his [6].


Theorem 6.1. All the embeddings

N ([u]) ⊂ N (Θ) ⊂ N ([u])〈Ψ±〉 ⊂ C∂(Θ(xU +D) , Θ(xU +D +U(P))eII(P)x)

are the Liouvillian extensions.

Proof. The fact that embedding N ([u]) ⊂ N (Θ) is Liouvillian follows directlyform formula (9). The statement concerning the second embedding is a consequenceof (13) because R([u];λ) ∈ N (Θ). Rewriting formula (6) in the form

Θ(xU +D +U(P))eII(P)x = Ψ+ ·Θ(xU +D) ,

we deduce that property for the last embedding to be Liouvillian results from theproportionality of (6) and expression (13) for Ψ+ (Theorems 4.2 and 5.1). �

Theorem 6.2. For generic λ integration of equation (1) in the Θ-representa-tion is equivalent to a multiplication of an element Ξ(x) generating the field N (Θ)by an adjoined linearly exponential divisor:

Proof. Consider Ξ(x) = Θ(xU +D)−1. It is clear that C∂(Ξ) = N (Θ). Then

(39) Ψ±(x;λ(P)) = C± · Ξ(x) ·Θ(xU +D ±U(P)

)e±II(P)x

because all the holomorphic/meromorphic integrals on hyperelliptic curves changesign under permutation of sheets μ → −μ; we may write±U(P), ±II(P) in (39). �

This theorem has an important treatment:

• When passing to the Θ-representation the equation (1) is integrated as if itwere an equation with constant coefficients. Integration procedure is thustrivialized under a proper choice of ‘domain of rationality’. The inversetransform method for the soliton class is a particular case of this generalconstruction.

Indeed, the simplest FG-case corresponds to the 0-gap one with N (Θ) = C(λ) andwe need only one linear exponent; the construction (39) acquires the form

(40) Ψ±(x;λ) = C± · Ξ(x) · e±a(λ)x , Ξ(x) = e0·x ∈ N (Θ) ,

wherein Ξ(x) has the ‘same form’ as the adjoint element ea(λ)x. Adjoining all theexponents associated with an N -soliton solution (and their varieties like positons(21) or rational solitons), we obtain the general 0-gap case.

Remark 6. The structure of solution (39) in form of simple multiplicationof elements generating N (Θ) and its extension N ([u])〈Ψ±〉 is not quite typicalfor equations integrable by attaching the linear exponents [34] or, especially, forequations with a solvable Galois group [31]. This property owes its origin to theavailability of λ in equation (Theorem 3.4).

Transition between u- and Θ-representations is transcendental with respectto the λ-dependence and other constants of the field. These constants are theΠ-matrices of curves, U -periods, and vector D. We may therefore trivialize thescheme above if we proceed further and redefine equation (1) over (37) as one givenover the λ-pencil (field) of the Lx(P)-divisors:

Lx(P) := Θ(xU +D +U(P))eII(P)x .

20 YU. BREZHNEV

Although field C∂

(Lx(P)

)is generated by the infinite number of elements, equation

(1) itself constitutes an infinite λ-pencil of equations. Strictly speaking, both spec-tral and quadrature approaches require to look upon Eq. (1) as being a differentially-algebraic one: differential in x and algebraic (polynomial) in λ. Furthermore, byvirtue of Proposition 5.5, the arbitrary Lx(P)-divisor generates some solution ofNovikov’s equation. It may be fixed by choice of one element of the pencil Lx(P):

(41) N (Θ)→ C∂(Θ(xU +Do +U(Po))eII(Po)x) .

Having extended the field (41), that is C∂

(Lx(Po)

), to the field C∂

(Lx(P)

), its

Galois group becomes trivial since the integral of equation (1) is given now in formof a ratio of two field elements.

7. Integrability and differential closedness

7.1. Differential closure in terms of θ-functions. Attaching the divisorLx(P) as transcendental element with a parameter P tells us that it should beintroduced to the theory as the base function, along with the available Θ’s withoutparameter P. Owing to Theorem 6.2 this would arrive us at a closed differentialapparatus (differential closedness) accompanying spectral equation and potential.Presently, the general Θ-formula realization of this viewpoint is an open problem,which is why we illustrate it by cases when g-dimensional Θ-function reduces to acombination of Jacobian ones.

At first glance, from Theorem 5.2, it would seem that supplement of the basis(32) with Lx-divisor of the type θ1(x − u)ehx requires also adjunction of all thefunctions θ′1, θ2,3,4(x− u). That no such complication takes place will be apparentfrom the following statement.

Theorem 7.1. For Weierstrassian curve (10) with modulus τ = ω′/ω onedefines an elliptic Lx(P)-divisor Λ by the formula

Λ(x; u|τ ) := θ1(x− u|τ ) exp(θ′1(u|τ)θ1(u|τ)

x+ hx), u /∈ Zτ + Z ,

where h is an extra parameter. Then the six functions Λ, θ′1, and θk (k = 1, 2, 3, 4)satisfy the closed autonomous system of ODEs over the field C

(η, ϑ2, θ(u)

):⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂θk∂x

=θ′1θ1

θk − πϑ2k ·

θnθmθ1

, n =8k − 28

3k − 10, m =

10k − 28

3k − 8

∂θ′1∂x

=θ′1

2

θ1− π2ϑ2

3ϑ24 ·

θ22θ1− 4

{η +

π2

12

(ϑ43 + ϑ4

4

)}·θ1

1

Λ

∂Λ

∂x=

θ′1θ1

+πϑ2

2

θ1(u)· θ

31(u) ·θ2θ3θ4 + θ2(u)θ3(u)θ4(u) ·θ31

θ1 ·(θ22(u) ·θ21 − θ21(u) ·θ22

) + h

.(42)

Motivation for the theorem lies in the fact that Abelian integrals, on the onehand, are representable in terms of theta-functions and, on the other hand, aredifferentially closed: the base integrals of 1st, 2nd, and 3rd kind form a differentialbasis. Indeed, derivatives of integrals are functions and any function is expressedthrough the two base ones (℘, ℘′) (generators of an elliptic functions field) whichare in turn integrals of exact meromorphic differentials of 2nd kind. However theta-function representation for a 3rd kind integral is still lacking in system (32).


Proof. Meromorphic functions are at hand since they are formed by θ-quo-tients. Derivatives of meromorphic functions are again meromorphic ones but dif-ferentiation of θk(x) generates θ′1(x). The quotient θ′1/θ1 is proportional to theWeierstrass ζ-function which in turn represents a meromorphic (i. e. 2nd kind) el-liptic integral [4]; it is alone as genus g = 1. Without loss of generality we may setthat the missing integral of 3rd kind has two logarithmic singularities. The place ofone of them may be fixed at x = 0 and the second place can be taken as a parame-ter. Call it u. Residues of a corresponding differential are opposite in sign and theycan be moved to a common multiplication constant. Such an integral III is unique(even for arbitrary g) up to a holomorphic one(s) since all the other integrals areexpressed through III = III(x; u). Its derivatives are again meromorphic functionsand the process closes. It will suffice to add an exponent of III and integral III itselfis given by the known formula

(43) III(x; u) :=1

2

z∫w + wo

z − zo

dz

w= ln

σ(2x− 2u)

σ(2x)e2ζ(2u)x

,

where z = ℘(2x), w = ℘′(2x) and zo = ℘(2u), wo = ℘′(2u). Weierstrassian pa-rameters (ω, ω′), presented in (43), are replaced by the one quantity τ thanks tohomogeneity relation ω2℘(ωx|ω, ω′) = ℘(x|1, τ ) =: ℘(x|τ ). Holomorphic integral isabsent in the system (32) but is present in a basis of Abelian integrals. Missingelement Λo can be formally adjoined by setting Λo(x|τ ) = x and supplementingsystem (42) with equation dΛo

dx = 1. Adding to (43) the holomorphic integral hΛo

and converting the right hand side of (43) to the θ-functions, one obtains thatadjunction of exp(III) is equivalent to adjunction of the object Λ(x; u|τ ). Differen-tiating (43) and converting it to the θ’s, one arrives, after some simplification, atthe last equality in Eqs. (42).

If u ∈ Zτ + Z then the integral III turns into a logarithm of meromorphicfunction: ln(℘(x)− e). There is nothing to adjoin. �

Corollary 7.2. Let potential be a finite-gap one and g-dimensional jacobiansplit into a product of the elliptic curves. Then differentiation of g-dimensionalfunctions Θ

(xU +D

)reduces to a set of the ‘1-dimensional’ equations (42). The

functions θk, θ′1 and Λ, taken possibly with different moduli τ , form the differentially

closed basis over which every Novikov’s equation of order � 2g+1 and problem (1)are integrated.

In the framework of this corollary integrability of Novikov’s equations is amanifestation of differential closedness of the first two equations in (42). The thirdequation in (42) ‘integrates’ equations with a parameter and their consequences(see examples in [12]). Here, ‘integrates’ means that all these equations are nothingmore than combinations of system (42) and its derivatives. Let us consider someexamples.

Example 4. The two gap Lame potential

Ψxx = (24℘(2x|τ ) + λ)Ψ .

22 YU. BREZHNEV

This is a classical example presented in many places [8, 27]. Corresponding solutionexpressed in terms of our objects reads as follows:

Ψ± =d

dx

Λ(x;±u|τ )θ1(x|τ )

, h =±2μ

3λ2 − 122g2(τ ), ℘(2u|τ ) = λ3 + 123g3(τ )

36λ2 − 123g2(τ ),

μ2 =(λ2 − 48g2(τ )

)(λ3 − 36g2(τ )λ+ 432g3(τ )

) (⇔ (8)

),

(44)

where g2(τ ), g3(τ ) are the standard modular τ -representations for Weierstrassianparameters a, b entering into the curve w2 = 4z3 − az − b [4]. Novikov’s equation(15) is satisfied under (c1, c2) = (0,−672g2(τ )) and R-polynomial has the form

R([u];λ) = λ2 − 1

2uλ+

1

4u2 − 36g2(τ ) ,

where u = 24℘(2x|τ ).

7.2. Non-elliptic example. We consider here a potential being no ellipticfunction but expressible through the ‘elliptic’ θ-functions.

Example 5. The Ψ-function for non-elliptic potential (35). It is a nonlinearsuperposition of the one-gap Ψ-functions. If we denote for brevity z = Ux+A andw = V x+B we then derive that

(45) Ψ(x;λ) =Λ(z+ 1

2 τ ;U1

∣∣τ)Λ(w+ 12 ;U2

∣∣κ)− Λ(z;U1|τ )Λ(w;U2|κ)θ1(z+ 1

2 τ∣∣τ)θ1(w+ 1

2

∣∣κ)− θ1(z|τ )θ1(w|κ)eII(λ)x

,

where holomorphic integrals U1, U2 are the certain linear combinations Uk = Ckjuj

of the elliptic holomorphic ones u1, u2 because the curve (8) corresponding to theΘ-function (34) has the form

(46) μ2 = λ(λ− 1)(λ− a)(λ− b)(λ− ab) =: P5(λ)

and is realized as covers of two tori defined by moduli τ and κ:

(47)

℘(2u1|τ ) + ϑ42(τ ) + ϑ3

3(τ ) = 3(1− a)(1− b)λ

(λ− a)(λ− b)ϑ43(τ ) ,

℘(2u2|κ) + ϑ42(κ) + ϑ3

3(κ) = 3(1− a)(1− b)λ

(λ− a)(λ− b)ϑ43(κ) .

These are the classical formulae by Jacobi [8] presented in terms of Weierstrass’ ℘and the pair of branch points (a, b) and moduli (τ,κ) are commonly written downfor one another [8]. All the information concerning this curve can be found in [8]and we omit details of some calculations. We need to compute the integral II(λ).

Abelian integrals for the curve (46) are expressed through θ, θ′1 and thereforederivation of the meromorphic integral II(λ) is a routine calculation. An explanationis that the reduction of holomorphic integrals to elliptic ones entails the reductionof the meromorphic Abelian integrals to the meromorphic elliptic ones. The latterare expressed through Weierstrassian ζ-function, i. e. θ′/θ, and meromorphic ellipticfunctions. First translate formulae for cover (47) into the language of θ-functions:

(48)ϑ22(τ )

ϑ23(τ )

θ24(u1|τ )θ21(u1|τ )

=(1− a)(1− b)λ

(λ− a)(λ− b),

ϑ22(τ )

ϑ23(τ )

θ24(u1|τ )θ21(u1|τ )

=ϑ22(κ)

ϑ23(κ)

θ24(u2|κ)θ21(u2|κ)

.

This is a complete set of equations determining the holomorphic integrals u1, u2 asfunctions of λ. The point λ =∞ corresponds to the values u1 = 1

2 τ and u2 = 12 κ.


Drop out for the moment indication of modulus τ in the following transformationof a meromorphic elliptic integral:∫

sds√4s3 − g2s− g3

=

∣∣∣∣s = π2

12

(ϑ43 + ϑ4

4 − 3ϑ23ϑ

24

θ23(u1)

θ24(u1)

)∣∣∣∣= −1

2

θ′1(u1− 1

2 τ)

θ1(u1− 1

2 τ) − 2ηu1 = · · ·

On the other hand, the 1st formula in (48) supplemented with use of the standardquadratic ϑ, θ-identities brings this integral into the following expression:

· · · = const

∫s · λ−

√a√b√

P5(λ)dλ �

∫ ((λ− a)(λ− b)

(1− a)(1− b)λ+

ϑ42 + ϑ4

3

3ϑ43

)λ−

√a√b√

P5(λ)dλ,

that is meromorphic integral on the curve (46) with a pole at λ =∞ and a surplusone at λ = 0. Doing the same for the second torus (u2) with modulus κ, weobtain one more meromorphic integral with the same infinities. Forming their linearcombination, we can construct the integral with a single singularity at infinite point.After some algebraic simplifications the sought-for result becomes:

(49)

II(λ) = a ·θ′1(u1− 1

2 τ∣∣τ)

θ1(u1− 1

2 τ∣∣τ) + b ·

θ′1(u2− 1

2 κ∣∣κ)

θ1(u2− 1

2 κ∣∣κ) + c ·u1 + d ·u2

= a · θ′1(u1|τ )θ1(u1|τ )

+ b · θ′1(u2|κ)θ1(u2|κ)

+(λ+ p)μ

λ(λ− a)(λ− b)+ q ·u1 + r ·u2 ,

where constants (a, b, c, d, p, q, r) depend only on parameters of the potential (35),i. e. on (τ , κ, A, B, V ), and are independent of λ. Again, after careful co-ordinationof all the moduli and normalizing constants the direct check of (1), (35), (45), and(49) becomes a good exercise in a differential θ-calculus. We mention in passingthat this example cannot be elaborated in the framework of the standard ellipticsoliton theory [8].

7.3. A new treatment of the spectral parameter. In proof of Theo-rem 7.1 parameter u was the only ‘external’ parameter independent of ‘internal’parameters of the curve (moduli). On the other hand, the only parameter beingexternal to the field N ([u]) and equation (1) is λ. It plays an isolated role. Apartfrom the fact that it is merely present in equation, it may be treated as an objectarising from the two independent ‘mechanisms’: 1) adjunction of a transcendentalelement and, on the other hand, 2) differential closedness of all the Abelian inte-grals. As evidenced by the foregoing and formula (25), these are the same things:

• The logarithmic singularity in a canonical integral of third kind is arbitraryand independent of moduli. The property of the theory to be integrable is,by construction, independent of it. It may therefore always be thought ofas a (free) spectral variable.

The converse is also true. Differentiation of the 3rd kind integrals yields otherintegrals and functions. In other words, informally speaking, one may say that

• Closed class of ODEs integrable through Θ(xU +D) is in fact integrablein terms of {Θ, Θ′, Lx} and ‘owes’ to contain an external (except formoduli) parameter P.

24 YU. BREZHNEV

Indeed, there is one fundamental logarithmic integral III for each algebraic curveand it has a single parameter P ⇔ λ. In the elliptic case this III has form (43) and forarbitrary genera it is expressed through the g-dimensional Θ’s [6]. The one fold log-arithmic ∂x-derivative of (25), i. e. derivatives of III(γ

′s;P), yields the rational func-tions of (z, w)(γ′s) and therefore only meromorphic objects remain since integralsthemselves disappear. As a rough guide we have here ∂x exp(III) = IIIx · exp(III).This linear in exp(III) equation9 is treated as a spectral one. Coefficients of thisequation (more precisely its λ-independent pieces) can be thought of as poten-tial. The quantity P should always be distinguished as an external one because,otherwise, the following chain

external λ ⇔ ∂x-closedness ⇔ Ψ-‘linearity’

is destroyed altogether. Moreover, the x-dependence Θ(xU +D) is not bound tobe a linear one and spectral equations must not necessarily be of the form (3).Counterexamples in Sect. 9 illustrate these points.

8. Definition of θ through Liouvillian extension

Insomuch as the object Λ(x; u|τ ) is in fact a theta-function with a parameter,we can use the differential equations described above as the basis for a definition ofthe theta’s themselves and, in particular, consider character of their integrability.

Proposition 8.1. The system (32) has the two algebraic (rational) integrals

ϑ22θ

24 − ϑ2

4θ22 = A1ϑ

23θ

21 , ϑ2

2θ23 − ϑ2

3θ22 = A2ϑ

24θ

21

generalizing the famous Jacobi θ-identities when A1 �= 1 �= A2.

Proof. The straightforward calculation shows that ∂xA1 = ∂xA2 = 0. �In a nutshell, the differential genesis of the object Λ is as follows. Function θ′1

is determined differentially through θ1. Therefore two functions with two arbitraryconstants solve the system (32). One of constants serves a homogeneity θ → Cθ of(32). Another one u is non-algebraic and is related to an autonomy of equations(32). Hence the two transcendental functions θ1(x − u|τ ) and θ2(x − u|τ ) remain.These functions are represented, up to a shift and holomorphic integral, by the oneobject Λ(x; u|τ ).

Theorem 8.2. Differential equations (42) are algebraically integrable.

Proof. By a direct computation one can show that any solution θk = θ of thesystem (42) satisfies the same 5th order ODE

(50)

(1

Fx(Fx

2

F )x

)x

+ 8Fx = 0 , F = (ln θ)xx − 2κ, −κ := 2η +1

6π2 (ϑ4

3 + ϑ44) .

Therefore, not taking into account u, there is only one essential parameter in equa-tions (42), i. e. parameter κ. From this it follows that

F = Ξ(x;a, b, c) :

F∫dz√

z(z − a)(z − b)= 2ix+ c

9An important remark is in order. We might not say the same as applied to the ‘pure spectral’object (27) since non-indefinite integral remains. Again, Weierstrass’ theorem does the job. The‘one-fold ∂x’, which is equivalent here to the ‘∂x-differential closedness’, explains why the spectralequations are always linear.


and integration is thus completed if, according to definition in Remark 3, we adjointhe inversion operation Ξ:

(51) θ = exp

x∫ { x∫Ξ(y;a, b, c)dy

}dx · eκx2+dx+e ,

where a, b, c,d, e are the integration constants. Of course, the inversion function Ξhere bears no relation to ratios of the θ-series. Integration of equations for functionsθ′1 and Λ is obvious. �

Remark 7. In a separate work we shall show that the algebraic integrabilityabove can be supplemented with a Hamiltonian formulation X = Ω∇H(X) to thesystem (42) and its Lagrangian description.

It is noteworthy, that variable θ satisfies an equation of fifth order, not third, asit would be expected from the well-known ℘-equation of Weierstrass [4]. Anotherpoint that should be mentioned here is the fact that algebraic integrability of equa-tions (42) leads not merely to the θ-function itself but to the elliptic Lx-divisor andeven its non-canonical extension by the quadratic exponent. (Notice that constantκ depends on modulus but constant d is free.) Further, the two-fold integration ofthe transcendental inversion operation in (51) can be reduced to integration of analgebraic function—our base operation.

Corollary 8.3. The θ-function can be defined through a meromorphic ellipticintegral.

To prove this it will suffice to make the following substitution in formula (51):

x∫Ξ(y;a, b, c)dy =

Ξ(x;a,b,c)∫zdz√

z(z − a)(z − b).

By this we obtain somewhat nonstandard way of introduction of the θ-functions.To all appearances, Tikhomandritskiı [52] was the first to point out a way of defini-tion of the θ through a meromorphic integral10 but his note [52] went unnoticed inthe literature. He poses a question about the natural going from elliptic integralsto the theta-functions and presents the mode of transition between these transcen-dents by introducing the integral of the 2nd kind elliptic integral. Indeed, rewritingformula (51) in the following form

(52) θ(x) = exp

x∫ { Ξ(x)∫zdz√

z(z − a)(z − b)

}dx · eκx2+dx+e ,

we observe that such a way of introduction of a θ-function is in effect the resultof Liouvillian extension of a meromorphic integral, i. e. adjoining an exponent ofintegral of such an integral. By this means we may adopt this point as a differ-ential definition of the θ a la Liouville and, subsequently, construct all the otherobjects of the theory: meromorphic (algebraic) functions are the θ-ratios, Abelianintegrals of 2nd kind are expressed through just introduced meromorphic integral(or, which is the same, the θ′), and the 3rd kind integrals are the logarithmic ratiosof the θ’s with free parameters (the Λ-objects). Holomorphic integrals are of coursethe independent objects; they are not defined/determined through any other ones.

10He does not mention the quadratic extension and differential closedness of the set {θk, θ′1},however.

26 YU. BREZHNEV

An important role of a meromorphic integral was already observed by Clebsch &Gordan in preface to their book [17, p. VI] on the base of Jacobi’s formula

Θ(u)

Θ(0)= exp

u∫0

Z(u)du,

where Z(u) is a Jacobi zeta-function notation in the theory of elliptic functions [4].The above differential properties of the Θ’s splittable to θ’s raise the question as

to whether the general multi-dimensional Θ-functions admit the similar ‘differentialkind’ definition. In particular, whether exists the purely Liouvillian definition ofan x-section Θ(xU + D) like formula (52)11 or, if any, the closed set of partialDEs defining the complete set of the general Θ, Θ′(z)-functions as ones of the garguments z? Some relations between Θ’s and meromorphic Abelian integrals canbe found in lectures by Weierstrass (though no really this point has been mentionedin the modern literature) but the question about closed and differentially Liouvilliandefinition (if it exists) of a 4g-set of the g-dimensional Θ-functions and associatedderivatives Θ′ remains an important open problem.

9. Non-finite-gap integrable counterexamples

Definition of integrability domain is not a subject of the Θ-function techniques.Therefore we may generate integrable equations by any way differed from the classi-cal FG-structure defined by formula (6) and Theorem 5.1. For example, equations

coming no from operators L({U}; ∂x) by taking the canonical spectral equation

L({U}; ∂x)Ψ = λΨ form in general the operator λ-pencils L({U}; ∂x, λ)Ψ = 0, say,(20). Their solution structure is not known a priori12. Moreover, we can evenconstruct an equation fitting no in the FG-scheme but having the same formalΘ-function form of solution.

Example 6. Omitting in notation the elliptic modulus τ , elucidate the said aboveby the following modification of the 2-gap Lame equation:

Ψxx ={24℘(2x) + 8℘(2x− u) + 16℘(u)

}Ψ .

It has a solution of the formal 2-gap Baker–Akhiezer form (44):

(53) Ψ(x; u) =d

dx

Λ(x; u)

θ1(x), h = 4ζ(u)− 2ζ(2u)

(exercise: check this solution). By Theorems 7.1 and 8.2 this example is alge-braically integrable over C∂

(℘(2x), ℘(2x− u)

)(and over C∂(θ1,Λ), of course) with

solvable Galois group but it has little in common with commutative Burchnall–Chaundy operators, BA-function, or spectral lacunae. Formula (53) shows that

11Roughly speaking, one needs an extensive strengthening of Theorem 6.1; whence it followsthat

−2 lnxxΘ(xU +D) = lnxxΨ+ (lnxΨ)2 + const

and, since the Ψ is an exponent of the 3rd kind Abelian integral (formula (25)), that isΨ = expIII(γ’s), the object Θ(xU + D) itself is computed as a ‘Liouvillian extension of Abelianintegrals’:

Θ(xU +D) = exp−1

2

{III(γ’s) +

∫∫ [III(γ’s)

]2xdxdx

}eax2+bx+c

which is a reminiscence of formula (51).12It is well known, however, that Eq. (20) is related to a matrix canonical eigenvalue problem.


this Ψ-function has no even a pole at point 2x = u where potential does. This poledepends on a ‘spectral parameter’ u.

Nevertheless this example should not be considered as ‘too artificial’ becausethere exist the Θ-function integrable models having algebraic curves with x-de-pendent branch-points. Ernst’s equations in general relativity [35] provide a niceexample along these lines.

9.1. Hermite’s operator pencil. Consider now, in the framework of Picard–Vessiot theory, Hermitian equation (11) itself. It is not a Burchnall–Chaundy op-erator but the operator λ-pencil. By virtue of formulae (11)–(13), we define thispencil over field N ([u]) and repeat arguments about λ-dependence of R.

One of solutions to this pencil is not exponential but a purely Abelian mero-morphic function (22), i. e. differential polynomial

R1 = R([u];λ) =:P � N ([u]) .

Since base of solutions to Eq. (11) is {Ψ+2 , Ψ+Ψ−, Ψ−

2 } and Ψ+2 /∈ N ([u]), we may

put the second solution as a square of the BA-function R2 = Ψ+2 , where Ψ+ is an

adjoint transcendent

Ψ+ := exp

x∫Px + 2μ

2Pdx.

The third solution is R3 = Ψ−2 and we obtain that

(54) R1 = P , R2 = Ψ+2 , R3 =

P2

Ψ+2 .

Hence rationality domain is the same as in Theorem 3.4, that is N ([u])〈Ψ+〉, andthis extension is a Liouvillian one of the transcendence degree 1 (we consider onlythe generic case λ �= Ej). We therefore can obtain the following result.

Theorem 9.1. Under the generic λ �= Ej the Galois group of Hermite’s equa-tion (11) defined over N ([u]) is connected and isomorphic to the diagonal groupG = Diag(1, α, α−1). Equations (11) is factorizable over field N ([u]).

Proof. As in proof of Theorem 3.4 we perform a linear transformation of thebasis {R1, R2, R3} and check invariance of the base differential relations betweenR’s. Let us take relations of the zero and 1st order in derivatives:

R1 = P , R2R3 = P2 , (R2)x =Px + 2μ

PR2 , (R3)x =

Px − 2μ

PR3 ,

which result from properties (54). Using condition μ �= 0, one easily derives thatadmissible transformations are

R1 → 1 ·R1 , R2 → α ·R2 , R3 → δ ·R3

and R2R3 → 1 · R2R3. Hence δ = α−1. Solutions {R1, R2, R3} are in generalnot algebraic functions, hence α is a free nonzero complex number and we do notneed further to analyze the remaining relations of second order in derivatives ofR’s (they will be automatically satisfied). This yields a connectivity of the groupand the matrix13 Diag(1, α, α−1). Factorizations of Eq. (11) are deducible by use of

13All this can also be seen from the fact that Galois group belongs to SL3(C) and transcen-dence degree of the extension is unity.

28 YU. BREZHNEV

Liouville’s scheme since we know solutions to equation. For example

∂xxx − 4(u+ λ)∂x − 2ux =

(∂x +

Px + 2μ

P

)∂x

(∂x −

Px + 2μ

P

)=

(∂x +

Px + 2μ

P

)(∂x −

2μ

P

)(∂x −

Px

P

).

These factorizations are not unique because equation has order 3. �

9.2. Inversion of non-holomorphic integrals. As a last counterexamplewe consider an equation that leads, on the one hand, to a nonstandard case ofthe inversion problem, and, on the other hand, to a nonlinear x-evolution in atheta-function argument. As we shall see, there is no essential difference between(quadrature) integrability schemes of this example and those of pure FG-potentials.This example was already pointed out as non-standard in [53].

Example 7. Let us consider the following spectral problem

(55) Ψxx =λ

v2Ψ ,

where v = v(x). As long as we have deal with invariant integration of (55) (Sect. 3),the theory has just non-essential modifications and we restrict ourselves to writingdown all its attributes in a form of references source for the simplest but nontrivialcase g = 1. We put

(56) R(x;λ) = vλ− φ(x) , φ(x) := 2a(x− b)(x− c) ,

where a, b, c ∈ C. Novikov’s equation is the equation v3vxxx −4φvx+4φxv = 0 andits integrals are as follows:

μ2 = λ3 + 3E2([v])λ2 + E1([v])λ+ a2(b− c)2 ,

3E2 = −1

2vvxx +

1

4vx2 − 2

φ

v, E1 =

1

2φvxx −

1

2φxvx +

φ2

v2+ 2av .

(57)

’Trace formula’ follows from a direct analogy of (22), i. e. we set R = (λ − γ)v,but inversion problem becomes an inversion of the logarithmic integral rather thanJacobi problem (26). Indeed, manipulations with integrals E1, E2 show that

(58)

γ∫1

z − E2

dz√4z3 − g2z − g3

=1

2a(b− c)ln

x− b

x− c+D,

where g2 = 12E22 − 4E1 and g3 = 4E1E2 − 8E2

3 − a2(b− c)2.From (58) it follows that the θ-function description undergoes changes since

x-evolution on jacobian is essentially non-linear. We pass from parameter E2 to� by the rule ℘(2�) = E2 and represent the logarithmic integral (58) in terms ofθ-functions of the holomorphic one r: γ = ℘(2r). We arrive at a transcendentalequation determining function r = r(x):

lnθ1(r − �)

θ1(r + �)+ 2

θ′1(�)

θ1(�)r =

℘′(2�)

a(b− c)ln

x− b

x− c+D.

As a result we obtain that ultimate solution to the Ψ-function is far from obvious:

Ψxx =λ℘2(2r)

a2(x− b)2(x− c)2Ψ , Ψ±(x;λ) =

√(x− b)(x− c)√

℘(2r)· Λ(r;±u)

θ1(r),


λ = ℘(2u)− ℘(2�) , h = 2ηu2 .

A direct check of this solution is a good exercise in theta-calculus. Moreover,nonlinearity of this x-evolution is two-fold. Logarithm (58) contains a fraction-linear function but the principal nonlinearity comes from a transcendental nonlin-earity of r(x). It never becomes linear even though we replace the general case in(56), that is φ(x) = 2a(x− b)(x− c), with the particular one φ(x) = const.

10. Concluding remarks

The properties of θ-functions outlined above differ in a crucial respect fromclassical special functions since the latter ones are defined by ODEs not integrablein quadratures over elementary or algebraic functions. For example Bessel’s func-tions or the Painleve transcendents. Therefore when generalizing rational theory(solitons), not only do algebraic integrability takes place for Novikov’s equationsbut it also takes place for linear spectral equations and even the θ-functions. Inall these cases the integration procedure has been closed at a single and commonstep: adjunction of the inversion operation Ξ. The elementary theory does not getby without inversion as well:∫

rational functions ⇒ ln ⇒ inversion ⇒ exponent ⇒ solitons .

It should be also emphasized that it makes no difference whether 1st kindintegrals (Jacobi problem) or 2nd, 3rd kind ones have been inverted (see (58)).The only thing is needed for the (Liouvillian) algebraic integrability: inversion ofindefinite integrals of any algebraic functions. Roughly speaking, the inversionprocedure appearing in ‘theta-methods’ has also the Liouvillian characterizationbecause, according to Eq. (16), adjunction of any kind Abelian integrals above isallowed. Nontrivial examples on inversion of meromorphic integrals can be found inmonograph [27]; they are associated with the Camassa–Holm hierarchy and havealso the theta-function description. In the same place quite extensive bibliographyis presented. In other words, in regard to invariant integrability, the choice of the Θ-series or the inversion operation Ξ is just a question of nomenclature. IntroducingΘ is equivalent to removing γ’s from formulae like (23)–(25) and conversely. Asfor analytic representation of solutions, the Θ-series is of course the fundamentalobject.

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E-mail address: [email protected]


Darboux transformation, exceptional orthogonal polynomialsand information theoretic measures of uncertainty

D. Dutta and P. Roy

Abstract. We study information theoretic measures of uncertainty for Dar-boux transformed partners of linear and radial harmonic oscillator. In par-ticular, we evaluate Shannon, Renyi and Fisher lengths for the isospectralpartner potentials whose solutions are given in terms of exceptional orthogo-nal polynomials and the results are compared with the corresponding ones ofthe harmonic oscillator.

Contents

1. Introduction2. A brief outline of supersymmetric quantum mechanics3. EOP’s associated with broken supersymmetry4. Relation with higher order Darboux transformation5. Information-theoretic lengths for the partner potential of linearharmonic oscillator via supersymmetry6. Moment of the quantum mechanical states associated with the partnerof harmonic oscillator via supersymmetry7. Discussion:References

1. Introduction

In quantum mechanics the number of exactly solvable potentials is few. As aconsequence it has always been of interest to find new exactly solvable potentials. Inthis regard factorization technique which was originally introduced by Schrodinger[1] plays a particularly important role [2]. Later it was shown [3] that factorizationmethod is closely related to Darboux transformation [4] and supersymmetric quan-tum mechanics [5]. Starting from a solvable potential Darboux transformation orsupersymmetric quantum mechanics can be used to construct new exactly solvablepotentials [6, 7, 8]. Here we shall follow the same procedure and apply Darbouxtransformation to the linear as well as radial harmonic oscillator potential. Howeverhere our objective is not to obtain new exactly solvable potentials but to examine

2010 Mathematics Subject Classification. Primary: 81Q60; Secondary: 33E30 .Key words and phrases. Exceptional orthogonal polynomials, measures of uncertainty, infor-

mation theory.


33

34 D. DUTTA AND P. ROY

the ways to use these solutions of the Darboux transformed potentials in the eval-uation of information theoretic measures of uncertainty. In this connection it maybe noted that Darboux transformation when applied in its simplest form does notproduce any interesting results as far as the nature of solutions are concerned i.e,the solutions are given in terms of classical orthogonal polynomials. However ifnon uniqueness property of factorization is used then Darboux transformation doesproduce very interesting results. In particular, the solutions of certain Darbouxtransformed potentials have been found to be in terms new types of orthogonalpolynomials rather than classical orthogonal polynomials.

Very recently two families of orthogonal polynomials related to the Laguerreand Jacobi polynomials, called the exceptional orthogonal polynomials (EOP) havebeen found within the Strum-Liouville framework [9]. These polynomials are dis-tinct from the classical orthogonal polynomials and consequently many of theircharacteristic properties are also different. It was also shown that EOP’s can alsobe obtained through Point Canonical Transformation method [10] as well as super-symmetric quantum mechanics (or Darboux-Crum transformation) [11, 12, 13].Subsequently EOP’s have been studied in a variety of contexts e.g, N fold super-symmetry [14], position dependent mass systems [15], shape invariance property[16], Dirac-Pauli-Fokker Planck equation [17] etc.

During the last few years various information theoretic measures of uncertainty[18, 19] have been studied by many authors. This is because these measuresare, in many ways, better suited than the Heisenberg measure of uncertainty [20,21]. However except for simple systems like the harmonic oscillator [19], Coulomb[22], Morse [23] or the Poschl-Teller potential [24] (whose solutions are given interms of classical orthogonal polynomials) [25] exact determination of informationentropies or other information theoretic measures of uncertainty is quite difficult.In a recent paper [26] we have studied Shannon entropy of various isospectralpartners of the harmonic oscillator whose solutions are given in terms of EOP’srelated to the Hermite and Laguerre polynomials. Here our objective is to evaluateother information theoretic measures of uncertainty like the Shannon length, Renyilength etc, for this class of problems. The organization of the paper is as follows : insection 2 we present outlines of supersymmetric quantum mechanics; in section 3 wepresent the main results [13] concerning EOP’s; in section 4 we explain the methodof obtaining EOP’s using higher order Darboux transformation; in sections 5 and 6various information theoretic measures corresponding to isospectral potentials areevaluated and finally section 7 is devoted to a conclusion.

2. A brief outline of supersymmetric quantum mechanics

Let us consider a pair of operators

(1) A+ =1√2

(d

dr+W (r)

), A− =

1√2

(− d

dr+W (r)

)With the help of these operators a pair of Hamiltonians can be constructed as [27]

(2) H± = A±A∓ =1

2

[− d2

dr2+ V±(r)

], V±(r) = W 2(r)±W ′(r)

The Hamiltonians H± form a supersymmetric pair and the function W (r) is calledthe superpotential. It is important to note that the Hamiltonians are positive

DARBOUX TRANSFORMATION, EXCEPTIONAL ORTHOGONAL POLYNOMIALS 35

semidefinite and obey the intertwining relations

(3) A+H− = H+A+, H−A

− = A−H+

As a consequence the above Hamiltonians are isospectral except perhaps the zeroenergy state (which if it exists is assumed to belong to H−). In this case super-symmetry is said to be unbroken and the relationship between the energies and theeigenfunctions of these Hamiltonians are given by

E−0 = 0, E−

n+1 = E+n > 0(4)

ψ−0 = N e−

∫W (r)dr, ψ+

n =1√E−

n+1

A+ψ−n+1, ψ−

n+1 =1√E+

n

A−ψ+n(5)

Thus the zero energy ground state is a singlet while the excited states are doublydegenerate. On the other hand if neither of ψ±

0 = e±∫W (r)dr is normalizable, then

supersymmetry is broken and we have [3, 27]

E+n = E−

n > 0(6)

ψ+n =

1√E−

n

A+ψ−n , ψ−

n =1√E+

n

A−ψ+n(7)

In this case all the states including the ground state are doubly degenerate. Fromthe relations (5) and (7) it follows that if solutions of one of the potentials areknown then the solutions of the other can also be found out.

3. EOP’s associated with broken supersymmetry

In this section we briefly describe properties of a class of EOP’s to be usedin later sections [13]. We consider the radial oscillator problem for which thesuperpotential is given by [29, 30]

(8) W (r) = r +γ + 1

r+

u′

u, 0 < r <∞

where u(r2) is given by

(9) u(r2) = 1F1

(1− ε

2, γ +

3

2,−r2

)It can be shown that neither of ψ±

0 = (u)±1r±(γ+1)e±r2

2 is normalizable so thatsupersymmetry is broken. In this case the partner potentials are found to be

V+(r) =r2

2+

γ(γ + 1)

2r2+ ε+ γ +

1

2,(10)

V−(r) =r2

2+

(γ + 1)(γ + 2)

2r2+

u′(r2)

u(r2)

(2r + 2

γ + 1

r+

u′(r2)

u(r2)

)− ε+ γ +

3

2,

(11)

Eq.(10) represents a radial oscillator potential with a shifted energy scale while (11)is a conditionally exactly solvable one [28] and has the same spectrum as V+. Theenergy values and wave function for V+(r) are given by(12)

E+n = 2n+ 2γ + 2 + ε, ψ+

n =

√2(n!)

Γ(n+ γ + 32 )

rγ+1Lγ+ 1

2n (r2) e−

r2

2 , n = 0, 1, 2, ....


On the other hand using the supersymmetric operator A− we can arrive at thesolution of V− as(13)

ψ−n (r) =

√2(n)!

(4n+ 4γ + 4 + 2ε)Γ(n+ γ + 32)

e−r2

2 rγ+2

u(r2)

[u′(r2)

rL

γ+ 12

n (r2) + 2u(r2)Lγ+ 3

2n (r2)

]which can be re-written in the form

(14) ψ−n (r) = Nn

√w(r2)pn(r

2)

where

(15)

Nn =

√2(n)!

(4n+ 4γ + 4 + 2ε)Γ(n+ γ + 32 )

,

pn(r2) =

[u′(r2)

rLγ+ 1

2n (r2) + 2u(r2)L

γ+ 32

n (r2)

],

w(r2) =e−r2r2γ+4

u2(r2)

If we choose the parameter ε in such a way that u(r2) is a polynomial, then{pn(r2)}∞n=0 forms a set of polynomials which are orthogonal with respect to theweight function w(r2). These polynomials are called EOP [13]. Below we presentsome properties of the EOP’s which are independent of a specific choice of γ or ε[13].

Orthogonality :

(16)

∫ ∞

0

w(r2)pm(r2)pn(r2)dr =

(2n+ 2γ + 2 + ε)Γ(n+ γ + 32 )

n!δmn

Generating function :

(17) F (r, z) =

∞∑n=0

pn(r2)zn =

er2zz−1

(1− z)γ+5/2

[(1− z)

u′(r2)

r+ 2u(r2)

]Differential equation for pn(r

2) :

(18) hpn(r2) = 0, h =

d2

dr2+ 2

√w

′√w

d

dr+

[√w

′′√w

+ 2(E−n − V−)

]

Ladder operators : For the polynomials pn(r2) there exists ladder operators L,L†

such that(19)

Lpn(r2) = −2(2n+ 2γ + 2 + ε)√(2n+ 2γ + 1)(2n+ 2γ + 3) pn−1(r

2),

L†pn(r2) = −2(n+ 1)(2n+ 2γ + 2 + ε)

√2n+ 2γ + 5

2n+ 2γ + 3pn+1(r

2)


where L and L† are given by(20)

L =

(− d

dr+W −

√w

′√w

)[(d

dr+ r +

√w

′√w

)2

− (γ + 1)(γ + 2)

r2

](d

dr+W +

√w

′√w

)

L† =

(− d

dr+W −

√w

′√w

)[(d

dr− r +

√w

′√w

)2

− (γ + 1)(γ + 2)

r2

](d

dr+W +

√w

′√w

)Rodrigues formula

(21) pn(r2) =

(−1

4

)n Γ(γ + 1 + ε2)

n!Γ(n+ γ + 1 + ε2)

√(γ + 3

2)

(n+ γ + 32)(L†)np0(r

2)

Polynomial algebras : Unlike the classical orthogonal polynomials, the EOP’s areassociated with nonlinear algebras rather than Lie algebras. Here it will be seenthat certain type of cubic algebra can be realized over space of EOP’s describedearlier. From (18) and (19) it can be shown that the operators L†, L and h satisfythe following commutation relations:

(22)[L†, h] = 4L†, [L, h] = −4L[L,L†] = −h[2(h+ 4γ + 2ε+ 4)2 − (h+ 4γ + 2ε+ 4)(2ε+ 10γ + 9)

+ 4γε+ 10ε+ 8γ2 + 36γ + 40]

Now from (9) it follows that u(r2) would be a polynomial only for positive oddintegral values of ε i.e, ε = 2m + 1, m = 1, 2, · · · . Till now we have not used anyparticular value of ε and consequently specific forms of either the potentials or thepolynomials are unknown. To have a look at their forms, we consider the simplestpossibility and take ε = 3. Then the partner potentials V± are given by

V+(r) =r2

2+

γ(γ + 1)

r2+ γ +

7

2(23)

V−(r) =r2

2+

(γ + 1)(γ + 2)

2r2− 4

2r2 + 2γ + 3+

16r2

(2r2 + 2γ + 3)2+ (2γ + 5)(24)

Some members of the family of EOP’s defined by Eq.(15) are given by

(25) p0(x) =1

(γ + 32 )

(2x+ 2γ + 5), p1(x) =1

(γ + 32 )

(2γ2 + 10γ +

21

2− 2x2

)It is to be noted that the EOP’s are considered in terms of the variable x = r2 ∈[0,∞).

3.1. EOP’s arising from systems on the whole real line. Here we shallbe considering EOP’s arising out of a system with unbroken supersymmetry. Suchpotentials can be obtained using either the radial oscillator or the linear oscillatorproblem [13]. Since in the previous section we have discussed a radial problem,here we shall consider an anharmonic oscillator problem with equidistant spectra [6]associated with the linear oscillator defined on the whole real line i.e. on (−∞,∞). The superpotential for this problem is defined by [29, 30]

(26) W (x) = x+u′

u−∞ < x <∞


where u(x) is given by

(27) u(x) = 1F1

(1− ε

2,1

2,−x2

)Proceeding as before one can obtain the partner potentials as

(28)V−(x) =

1

2x2 +

u′

u

(2x+

u′

u

)− ε+

1

2V+(x) = 1

2x2 + ε− 1

2

It is seen that while the potential V+(x) is essentially a linear oscillator its partnerV−(x) is a completely different potential. One can easily verify that in this casesupersymmetry is unbroken so that the ground state of V−(x) is a singlet. Further-more using the solutions of V+(x) one can obtain the solutions of V−(x) and theyare given by [29](29)

E−0 = 0, ψ−

0 ≈ e−x2/2

u

E−n+1 = n+ ε, ψ−

n+1(x) =1√

2n+1n!(n+ 3)√π

√w(x)qn+1(x), n = 0, 1, 2, ....

where

(30) q0(x) = 1, qn+1(x) = u(x)Hn+1(x) + u′(x)Hn(x), w(x) = e−x2

u2(x)

There are a few points to be noted. First, unlike the previous case here the poly-nomial family starts with a constant. Secondly, the subsequent members of thefamily can not be obtained from the first member because the ground state is asinglet and it is annihilated by both the operators A±. Finally, as a consequencean algebra like (22) can not be realized over the space of all the polynomials butonly on the set {qn+1(x)}∞n=0 [13, 29].

4. Relation with higher order Darboux transformation

So far we have considered exactly solvable potentials which are related by firstorder supersymmetry. On the other hand solutions of some potentials constructedusing higher order intertwining operators have recently been obtained in terms ofextended X1 type Jacobi polynomials [11]. We would like to show that some of theEOP’s considered arise as solutions to potentials constructed using second orderDarboux transformation. To show this let us first recall that if V (x) is a certainpotential then the potential V2(x) constructed as [7, 8, 31]

(31) V2(x) = V (x)− 2d2

dx2logWj,j+1,....j+m(x)

has the same spectrum as V (x) except the levels Ej , Ej+1,....,Ej+m. HereWj,j+1,....j+m(x) denotes the wronskian of the m consecutive levels starting withthe j th level.


Let us now consider ε = 3. Then the potential V−(x) in (28) can be found tobe [29]

(32)

H− = −1

2

d2

dx2+ V−(x)

= −1

2

d2

dx2+

x2

2− 4

1 + 2x2+

16x2

(1 + 2x2)2+

3

2

E−0 = 0, E−

n+1 = n+ 3, n = 0, 1, 2 · · ·whereas Hamiltonian for the linear harmonic oscillator is defined by

(33) H = −1

2

d2

dx2+

x2

2− 1

2, En = n, n = 0, 1, 2, ....

Comparing the spectrum of the above Hamiltonians we find that H− has exactlythe same spectrum as H except that the first and second excited states are missing.Now applying the second order Darboux algorithm (31) to the potential (33) forthose missing states (so here j = 1,m = 1), we arrive at V−(x) of (28). Let us nowconsider ε = 5 in (28) and obtain

(34)V−(x) =

1

2x2 +

u′

u

(2x+

u′

u

)− 9

2E−

0 = 0, E−n+1 = n+ 5, n = 0, 1, 2, · · ·

where u(x) = 1F1

(−2, 1

2,−x2

). It is obvious that the spectrum in this case is

the same as in (33) except that the first, second, third and the fourth excitedstates are deleted. It can be verified that on applying the fourth order Darbouxtransformation to (33) for those four missing states ( so choosing j = 1) we arrive at(34). In other words, by applying higher order Darboux transformation one obtainspotentials whose solutions are given by EOP’s.

However in the case of broken supersymmetry the situation is somewhat dif-ferent. Comparing the spectrum of the radial harmonic oscillator with that of thepotential V+(x) in (23) we find that V±(x) has one state missing and consequently ifwe apply the first order Darboux transformation we arrive at V+(x). But because ofbroken supersymmetry V± have identical spectrum and consequently higher orderDarboux transformation can not be applied to obtain V−(x).

5. Information-theoretic lengths for the partner potential of linearharmonic oscillator via supersymmetry

We would like to note that the information entropies like Shannon entropyor the Renyi entropy [18] are useful as a measure of uncertainty and they havebeen evaluated for some of the potentials considered here [26]. However they arenot direct measures of spreading of probability density in the sense that they donot have the same unit as x [32]. As a measure of uncertainty the information-theoretic lengths are physically much more appropriate than the standard deviationas these lengths are realized in the support interval, and not specified at a point.The Heisenberg measure for the quantum probability density ρn(x) characterized bythe quantum number is not generally an useful measure of uncertainty since insteadof measuring the extent to which the distribution is concentrated, it only measures


the concentration of the probability cloud around a particular point of support in-terval of density [22]. To overcome these consideration, use of information-theoreticbased uncertainty measures like Renyi, Shannon and Fisher lengths etc. have beensuggested by many authors [23, 25, 33].

In the evaluation of information theoretic lengths it is necessary to first eval-uate the spreading lengths of the solutions of Schrodinger equation. In turn thisrequires, at least in most cases, determination of the spreading lengths of variousclassical orthogonal polynomials associated with the solutions of standard problemsof quantum mechanics. Here our purpose is to determine the spreading lengths ofthe solutions of isospectral potentials obtained in earlier sections. In particularit will be seen that the results concerning the information theoretic measures (orspreading lengths) allow one to discuss and quantify the internal disorder of thesystem for both the ground as well as the excited states [34, 35].

5.1. Shannon length for the partner of harmonic oscillator potential.Like the Heisenberg uncertainty measure, the most well-known direct measure ofspreading is the Shannon length which is a kind of spreading measure of globalcharacter because of the fact that this length is associated with power like or log-arithmic functional of probability density. In other words Shannon lengths aremeasures of the extent to which the density is concentrated. The Shannon lengthis denoted by N [ρ] and is defined as [25, 33, 23]

(35) N [ρ] = eS[ρ]

where S[ρ] is the Shannon entropy and is defined as

(36) S[ρ] = −∫|ψ(x)|2log|ψ(x)|2 dx

Let us consider ε = 3. The the potentials in (28) and the corresponding eigenfunc-tions read [13]

V+(x) =x2

2+

5

2,

E+n = n+ 3,

ψ+n =

√1

2nn!√π

e−x2/2Hn(x)

(37)

V−(x) =x2

2− 4

1 + 2x2+

16x2

(1 + 2x2)2+

3

2E−

0 = 0, E−n+1 = E+

n = n+ 3,

ψ−n+1 =

1√E+

n

A−ψ+n

=1√

2n+1n!(n+ 3)√π

e−x2/2

(1 + 2x2)

[(1 + 2x2)Hn+1 + 4xHn

], n = 0, 1, 2, ...

(38)

The Shannon length for the nth state of V−(x) in (37) is given by

(39) N [ρ−n+1] = eS[ρ−n+1]


Here S[ρ−n+1] denotes the Shannon entropy of that (n+ 1)th state and is given by(40)

S[ρ−n+1] =K2

N2S[ρ+n+1]−

K2

N2

∫ ∞

−∞

[16x2H2

ne−x2

(1 + 2x2)2+

8xHnHn+1e−x2

1 + 2x2

]log

(e−x2

H2n+1

K2

)dx

− 1

N2

∫ ∞

−∞e−x2

[H2

n+1 +8xHnHn+1

1 + 2x2+

16x2H2n

(1 + 2x2)2

]

log

[K2

N2+

8K2xHn

N2(1 + 2x2)Hn+1+

16K2x2H2n

N2H2n+1(1 + 2x2)2

]dx

where K and N are given by

(41) K =

√2n+1(n+ 1)!

√π, N =

√2n+1n!(n+ 3)

√π

It may be noted that the Shannon lengths in (39) can be expressed partly interms of the Shannon lengths S[ρ+n+1] of the partner potential V+(x) which areknown [25]. Now using the intertwining relations (3) it can be shown from (40)that

(42) N [ρ−n+1] = eS[ρ−n+1] = e−T+K2

N2 S[ρ+n+1], n = 0, 1 · · ·

where

(43)

T =K2

N2

∫ ∞

−∞

[16x2H2

ne−x2

(1 + 2x2)2+

8xHnHn+1e−x2

1 + 2x2

]log

(e−x2

H2n+1

K2

)dx

+1

N2

∫ ∞

−∞e−x2

[H2

n+1 +8xHnHn+1

1 + 2x2+

16x2H2n

(1 + 2x2)2

]

log

[K2

N2+

8K2xHn

N2(1 + 2x2)Hn+1+

16K2x2H2n

N2H2n+1(1 + 2x2)2

]dx

We would like to note that since the ground state is a singlet, it must be treatedseparately and the Shannon length for the ground state of the V−(x) in (38) isfound to be

(44)N [ρ−0 ] = eS[ρ−

0 ] = e−∫∞−∞

2e−x2

√π(1 + 2x2)2

log[2e−x2

√π(1 + 2x2)2

]

In Table 1 we have given the results for Shannon length and Heisenberg uncertaintymeasure for the linear oscillator potential (37) and its partner (38) for a few levels.In both the cases it is found that the Shannon length as well the Heisenberg un-certainty measures increases with increasing n. Also, the Shannon length for thepartner potential is less than that of the oscillator. The same conclusion holds forthe Heisenberg uncertainty measure.

5.2. Renyi length for the partner of harmonic oscillator potential.Renyi length is one of the most important information-theoretic global direct spread-ing measure like Shannon length i.e, it also measures the extent to which the prob-ability density is concentrated. The general q th order Renyi length £R

q [ρn+1] isdefined as [23, 25, 33]

(45) £Rq [ρn+1] = eRq [ρn+1], q > 0, q �= 1


where

(46) Rq[ρn+1] =1

1− qlog

⟨[ρn+1]

q−1⟩

denotes the Renyi entropy.The second order Renyi length for the n-th state of the potential V−(x) in (38)

is given by

(47) £R2 [ρ

−n+1] = eR2[ρ

−n+1]

where R2[ρ−n+1] is the Renyi entropy of that particular state. As before it can be

expressed in terms of Renyi entropy of the linear harmonic oscillator in the followingway(48)

R2[ρ−n+1] = R2[ρ

+n+1]

− logK4

N4

∫ ∞

−∞

[1 +

256x4H4n

H4n+1(1 + 2x2)4

+96x2H2

n

(1 + 2x2)2H2n+1

+256x3H3

n

(1 + 2x2)3H3n+1

+16xHn

(1 + 2x2)Hn+1

]dx

where K and N are given by(41).Therefore the second order Renyi length (q = 2) of the state ψ−

n+1 is(49)

£R2 [ρ−

n+1] = exp(R2[ρ−n+1]) = exp(− log

⟨[ρ−

n+1]⟩) =

1∫∞−∞(|ψ−

n+1|2)2dx=

1∫∞−∞ | 1√

E+n

A−ψ+n |4dx

=eR2[ρ

+n+1

]

K4

N4

∫ ∞

−∞

[1 +

256x4H4n

H4n+1(1 + 2x2)4

+96x2H2

n

(1 + 2x2)2H2n+1

+256x3H3

n

(1 + 2x2)3H3n+1

+16xHn

(1 + 2x2)Hn+1

]dx

,

n = 0, 1, 2 · · ·

where R2[ρ+n+1] is already known [25]. Renyi length for the ground state of the

partner of linear harmonic oscillator potential is

(50)1∫∞

−∞ |ψ−0 (x)|4dx

=π

4

1∫∞−∞

e−2x2

(1+2x2)4 dx=

12√2π

22√π + πe erfc[1]

where ψ−0 (x) is obtained by solving the equation A+ψ−

0 (x) = 0. It is worth men-tioning that the Shannon length is the limiting case of Renyi length i.e. N [ρ] :=

limq→1

£Rq [ρ] = eS[ρ].

In Table 2 we have given the results of computation for Renyi length andHeisenberg uncertainty measure. From the Table 2 we find similar behaviour asthe Shannon length. However, it may also be seen that Shannon length for thepartner is always more than the Renyi length of the oscillator.

5.3. Fisher length for the partner of harmonic oscillator potential.There is yet another direct measure of spreading, namely the Fisher length, definedby the relation [25, 33, 23]

(51) (δx) :=1√F [ρ]

where F [ρ] denotes the Fisher information of the density ρ(x) and is defined as

(52) F [ρ] :=

⟨[d

dxln ρ(x)

]2⟩=

∫ ∞

−∞

[ρ′(x)]2

ρ(x)dx.


Since the Fisher length is a functional of the derivative of the density, it is a localmeasure of uncertainty and qualitatively different from the previous ones. In otherwords the Fisher length gives the degree of local disorder [22]. It is also interestingto note that Fisher information and the variance (see Section 6) satisfy the Cramer-Rao uncertainty relation [23]

(53) F [ρ](Δx)2 ≥ 1

Now for the state ψ−n+1(x) of the partner potential of linear oscillator (37), the

Fisher length is defined as

(54) (δx)−n+1 =1√

F [ρ−n+1]

where(55)

F [ρ−n+1] =

∫ ∞

−∞[d

dxρ−n+1]

2 dx

ρ−n+1

=

∫ ∞

−∞[d

dx|ψ−

n+1|2]2dx

|ψ−n+1|2

=

∫ ∞

−∞[d

dx| 1√

E+n

A−ψ+n |2]2 dx

| 1√E+

n

A−ψ+n |2

=4K2

N2F [ρ+n+1] +

4

N2

∫ ∞

−∞e−x2

(4xHnHn+1 − 4(2n+ 1)H2n)dx

+4

N2

∫ ∞

−∞

e−x2

(1 + 2x2)4{64n2x2(1 + 2x2)2H2

n−1 + (6− 4x2)[(6− 4x2) + 4n(1 + 2x2)2]H2n+

16nx(1 + 2x2)[6− 4x2 + 2n(1 + 2x2)2]HnHn−1 − 2x(1 + 2x2)2(6− 4x2)HnHn+1

−16nx2(1 + 2x2)3Hn+1Hn−1

}dx

where K and N are given by (41) and F [ρ+n+1] can be obtained from [25].Fisher length for the ground state of the partner of linear harmonic oscillator

potential is(56)

(δx)−0 =1√

F [ρ−0 ]=

1√∫∞−∞

[ddx

|ψ−0 (x)|2

]2dx

|ψ−0 (x)|2

=1√

2√π

∫∞−∞[ d

dxe−x2

(1+2x2)2]2(1 + 2x2)2ex

2dx

=

√3

2(9 +√2eπ erfc[ 1√

2])

we have computed the Fisher lengths for a number of levels of the partner potentialas well as the oscillator and the results are given in Table 3. It is seen that Fisherlengths for the partner is always less than those of the oscillator. Also it can beverified that the Cramer-Rao uncertainty relation is always satisfied as it shouldbe.

5.4. Shannon length for the partner of radial oscillator potential asso-ciated with the broken supersymmetry. Here we shall consider the potentialrelated to the radial oscillator potential. Unlike the previous examples, the presentone arises from a system with broken supersymmetry. The Shannon length for


the n-th state reduced wave function of effective potential V−(r) ( ε = 3 and γ isreplaced by angular momentum l ) in (11) is defined by

(57) N [ρ−n,l+1] = eS[ρ−

n,l+1]

where S[ρ−n,l+1] is the Shannon entropy of that particular state (13) and is given by

(58)

S[ρ−n,l+1] =4N2

n

KS[ρ+n,l+1]

−16N2n

∫ ∞

0

e−r2r2l+4

(2r2 + 2l + 3)2

(L

l+ 12

n (r2)2 + (2r2 + 2l + 3)Ll+ 1

2n (r2)L

l+ 32

n (r2)

)

log[ke−r2r2l+4Ll+ 3

2n (r2)2]dr

−N2n

∫ ∞

0

{ e−r2r2l+4

(2r2 + 2l + 3)2[4(2r2 + 2l + 3)2L

l+ 32

n (r2)2 + 16(2r2 + 2l + 3)

Ll+ 1

2n (r2)L

l+ 32

n (r2) + 16Ll+ 1

2n (r2)2]

log[N2

n

K(4 +

16Ll+1/2n (r2)2

(2r2 + 2l + 3)2Ll+3/2n (r2)2

+16L

l+1/2n (r2)

(2r2 + 2l + 3)Ll+3/2n (r2)

)]}dr, n = 0, 1, 2 · · ·

where S[ρ+n,l] is the Shannon entropy of (12) which is known [26] for ε = 3 and

(59) K =2(n!)

Γ(n+ l + 5/2), Nn =

√n!

(2n+ 2l + 5)Γ(n+ l + 3/2)

Now using (58) the Shannon length for the partner of radial harmonic oscillator isfound to be

(60)N [ρ−n,l+1] = e

S[ρ−n,l+1] = e

4N2n

KS[ρ+n,l+1]− T

where(61)

T = 16N2n

∫ ∞

0

e−r2r2l+4

(2r2 + 2l + 3)2

(L

l+ 12

n (r2)2 + (2r2 + 2l + 3)Ll+ 1

2n (r2)L

l+ 32

n (r2)

)

log[ke−r2r2l+4Ll+ 3

2n (r2)2]dr

+N2n

∫ ∞

0

{ e−r2r2l+4

(2r2 + 2l + 3)2[4(2r2 + 2l + 3)2L

l+ 32

n (r2)2 + 16(2r2 + 2l + 3)

Ll+ 1

2n (r2)L

l+ 32

n (r2) + 16Ll+ 1

2n (r2)2]

log[N2

n

K(4 +

16Ll+1/2n (r2)2

(2r2 + 2l + 3)2Ll+3/2n (r2)2

+16L

l+1/2n (r2)

(2r2 + 2l + 3)Ll+3/2n (r2)

)]}dr, n = 0, 1, 2 · · ·

It may be observed that the radial part of V+(r) is that of a standard radialoscillator while that of V−(r) actually depends on l. Another interesting point isthat in the present case the Shannon length for the ground state need not haveto be computed separately because as supersymmetry is broken it can be obtained


from that of the oscillator by using intertwining operator. We have evaluated theShannon lengths for the partner potential and the results of this computation aregiven in Table 4. From Table 4, it is seen that the Shannon length for different levelsfor the partner potential are more than those of the oscillator potential althoughl = 0 for the oscillator problem implies l = 1 for the partner potential.

6. Moment of the quantum mechanical states associated with thepartner of harmonic oscillator via supersymmetry

In most of the standard quantum mechanical models Heisenberg uncertaintymeasure behaves as a weaker measure of uncertainty but it is still of some interestto examine the strength of this spreading measure compared to the other ones. Inorder to find Heisenberg uncertainty measure for the solution of the Schrodingerequation it is necessary to evaluate the moments of the states. The general k-thorder moment for the isospectral partner of harmonic oscillator is

(62)⟨xk⟩−n+1

=

∫ ∞

−∞xkρ−n+1(x)dx =

∫ ∞

−∞xk|ψ−

n+1(x)|2dx

=

∫ ∞

−∞xk|

1√E+

n

A−ψ

+n (x)|2dx =

1

2n+1n!(n + 3)√π

∫ ∞

−∞xk[(1 + 2x

2)Hn+1 + 4xHn]

2 e−x2

(1 + 2x2)2dx

=1

2n+1n!(n + 3)√π

⎡⎣∫ ∞

−∞xkH

2n+1e

−x2dx + 8

∫ ∞

−∞xk+1

HnHn+1e−x2

1 + 2x2dx + 16

∫ ∞

−∞xk+2

H2n

e−x2

(1 + 2x2)2dx

⎤⎦

=1

(n + 3)

⎡⎣(n + 1)⟨xk⟩+

n+1+

8

2n+1n!√π

⎛⎝∫ ∞

−∞xk+1

HnHn+1e−x2

1 + 2x2dx + 2

∫ ∞

−∞xk+2

H2n

e−x2

(1 + 2x2)2dx

⎞⎠⎤⎦Hence the second order moment is

(63)⟨x2

⟩−n+1

=1

(n + 3)

⎡⎣(n + 1)⟨x2⟩+

n+1+

8

2n+1n!√

π

⎛⎝∫ ∞

−∞x3HnHn+1

e−x2

1 + 2x2dx + 2

∫ ∞

−∞x4H

2n

e−x2

(1 + 2x2)2dx

⎞⎠⎤⎦=

1

2n+1n!(n + 3)√π

[(2n + 3)2

n√π(n + 1)! + 2

n+2(n + 1)!

√π

−4∫∞−∞ xHnHn+1

e−x2

1+2x2 dx + 2n+2n!√

π − 8∫∞−∞ H2

ne−x2

1+2x2 dx + 4∫∞−∞ H2

ne−x2

(1+2x2)2dx

]

and the 1-st order moment is

(64)

〈x〉−n+1 =1

N2

∫ ∞

−∞x|ψ−

n+1(x)|2dx

=1

N2

∫ ∞

−∞x[(1 + 2x2)Hn+1 + 4xHn]

2 e−x2

(1 + 2x2)2dx = 0

as the integrand is odd function. So the position space Heisenberg uncertaintymeasure is given by the root-mean square or standard deviation which is a directspreading measure :

(65) (Δx)−n+1 =√〈x2〉−n+1 − (〈x〉−n+1)

2 =√〈x2〉−n+1

Making use of (63) one can compute the standard deviation (and hence the Heisen-berg uncertainty measure) for the bound state solutions of V−(x).


n N [ρ+n ] for Harmonic N [ρ−n ] for Heisenberg uncertainty Heisenberg uncertaintyoscillator(H.O) partner of H.O measure for H.O measure for partner of H.O

0 2.921 1.614 0.71 0.394

1 3.829 3.529 1.22 1.031

2 4.475 4.158 1.58 1.526

3 4.997 4.845 1.87 1.837

Table 1. Shannon lengths and Heisenberg uncertainty measuresfor linear Harmonic oscillator (37) and its partner (38).

n £R2 [ρ+n ] of Harmonic £R

2 [ρ−n ] for Heisenberg uncertainty Heisenberg uncertainty

oscillator(H.O) partner of H.O measure for H.O measure for partner of H.O

0 2.51 1.322 0.71 0.394

1 3.342 3.062 1.22 1.031

2 3.913 3.557 1.58 1.526

3 4.365 4.191 1.87 1.837

Table 2. Renyi lengths and Heisenberg uncertainty measures forlinear Harmonic oscillator (37 and its partner (38).

n (δx)+n for Harmonic (δx)−n for Heisenberg uncertainty Heisenberg uncertaintyoscillator(H.O) partner of H.O measure for H.O measure for partner of H.O

0 1√2

0.381 1√2

0.394

1 1√6

0.332√

32

1.031

2 1√10

0.199√

52

1.526

3 1√14

0.072√

72

1.837

Table 3. Fisher lengths and Heisenberg uncertainty measures forlinear Harmonic oscillator and its partner.

n N [ρ+n,l] for radial N [ρ−n,l+1] for partner of

oscillator radial oscillator0 1.915 1.9611 2.501 2.5542 2.929 2.9843 3.283 3.336

Table 4. Shannon lengths for radial oscillator (23) andits isospectral partner potential (24 for l = 0.

7. Discussion:

In this paper we have considered a class EOP’s associated with quantum me-chanical systems which exhibits unbroken as well as broken supersymmetry. Sub-sequently different information theoretic measures of uncertainty like the Shannonlength, Renyi length etc. have been evaluated and compared for some low lyinglevels of the harmonic oscillator as well as its isospectral partner whose solutionsare given in terms of EOP’s. In all the cases considered here it was possible toexpress various lengths partly in terms of lengths of harmonic oscillator potentialwhich are known and this made it possible to simplify the evaluation to some ex-tent. We would like to point out another feature of the results obtained here. Inthe usual cases one evaluates various information theoretic measures of uncertaintyfor a particular potential and then the behaviour of such measures can be studiedby varying some parameter(s) of the potential. In the present case, however, we


have compared different measures of uncertainty for a pair of potentials relatedby supersymmetry. More precisely, in the case of broken supersymmetry we havecompared uncertainties of two states having the same energy and quantum num-ber. On the other hand, in the case of unbroken supersymmetry we have compareduncertainty measures for two states which share one quantum number. Also, fromthe tables different uncertainty measures for the same (partner) potentials may becompared. We feel it would be interesting to compare measures of uncertainty ofstates with respect to other parameters. It may also be noted that we have eval-uated different measures of uncertainty for low lying levels only and it would beof interest to obtain these measures for large values of the quantum numbers byevaluating asymptotic forms of the EOP’s.


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Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-, 700108,

India.


Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-, 700108,

India.



On orthogonal polynomials spanning a non-standard flag

David Gomez-Ullate, Niky Kamran, and Robert Milson

Abstract. We survey some recent developments in the theory of orthogonalpolynomials defined by differential equations. The key finding is that there ex-ist orthogonal polynomials defined by 2nd order differential equations that falloutside the classical families of Jacobi, Laguerre, and Hermite polynomials.Unlike the classical families, these new examples, called exceptional orthog-onal polynomials, feature non-standard polynomial flags; the lowest degreepolynomial has degree m > 0. In this paper we review the classification ofcodimension m = 1 exceptional polynomials, and give a novel, compact proof

of the fundamental classification theorem for codimension 1 polynomial flags.As well, we describe the mechanism or rational factorizations of 2nd orderoperators as the analogue of the Darboux transformation in this context. Wefinish with the example of higher codimension generalization of Jacobi polyno-mials and perform the complete analysis of parameter values for which thesefamilies have non-singular weights.

Contents

1. Introduction2. Preliminaries3. Codimension 1 flags4. Higher codimension flagsAcknowledgementsReferences

1. Introduction

Even though the orthogonal polynomials of Hermite, Laguerre and Jacobi arosefrom various applications in applied mathematics and physics, these three familiesnow serve as the foundational examples of orthogonal polynomial theory. As such,these classical examples admit many interesting and fruitful generalizations.

A key property of classical orthogonal polynomials is the fact that they can bedefined by means of a Sturm-Liouville problem. One of the foundational resultsin this area a Theorem of Bochner [5] which states that if an infinite sequence of

2000 Mathematics Subject Classification. Primary 34L10, 42C05; Secondary 33C45, 34B24.Key words and phrases. Orthogonal polynomials; Sturm–Liouville problems; Exceptional

polynomial subspaces.


51

52 DAVID GOMEZ-ULLATE, NIKY KAMRAN, AND ROBERT MILSON

polynomials of degree 0, 1, 2, . . . satisfies a second order eigenvalue equation of theform

(1) p(x)y′′ + q(x)y′ + r(x)y = λy

then p(x), q(x) and r(x) must be polynomials of degree at most 2, 1 and 0 respec-tively. In addition, if the eigenpolynomial sequence is L2-orthogonal relative to ameasure with finite moments, then it has to be, up to an affine transformation ofx, one of the three classical classical families listed above [1, 29, 12, 27, 26]

Implicit in the above definition of classical polynomials is the assumption thatorthogonal polynomials form a basis for the standard polynomial flag P0 ⊂ P1 ⊂P2 ⊂ . . ., where Pn = span{1, x, x2, . . . , xn}. In a pair of recent papers [20, 21],we showed that there exist complete orthogonal polynomial systems defined bySturm-Liouville problems that extend beyond the above classical examples. Whatdistinguishes our hypotheses from those made by Bochner is that the eigenfunctioncorresponding to the lowest eigenpolynomial of the sequence need not be of degreezero, even though the full set of eigenfunctions still forms a basis of the weightedL2 space.

Already some 20 years ago, particular examples of Hermite-like polynomialswith non-standard flags were described in the context of supersymmetric quantummechanics [9, 6]. The last few years have seen a great deal of activity in thearea of non-standard flags; the topic now goes under the rubric of exceptionalorthogonal polynomials. There are applications to shape-invariant potentials [30],supersymmetric transformations [22], to discrete quantum mechanics [36], mass-dependent potentials [28], and to quasi-exact solvability [42]. As well, there arenow examples of orthogonal polynomials that are solutions of 2nd order equationsand that form flags of arbitrarily high codimension [35].

In light of the activity surrounding orthogonal polynomials with non-standardflags, we hope that it will be useful to summarize some key results and findings, andto supply stand-alone proofs to some key propositions. We note that the adjective“exceptional” was introduced in the context of our investigation of the equivalenceproblem for polynomial subspaces [20, 7].

2. Preliminaries

2.1. Polynomial flags. Let U1 ⊂ U2 ⊂ . . ., where dimUk = k be a flag ofreal, finite-dimensional, polynomial subspaces. Let nk denote the degree of Uk; thatis nk is the maximum of the degrees of the polynomials p ∈ Uk. Let �k = nk+1−kdenote the codimension of Uk in Pnk

, where the n+1 dimensional polynomial vectorspaces

Pn = span{1, x, x2, . . . , xn}make up the standard polynomial flag: P0 ⊂ P1 ⊂ P2 ⊂ . . .. We say that {pk}∞k=1

is a basis of the flag if

(2) Uk = span{p1, . . . , pk}Note that no generality is lost if we assume that deg pk = nk.

Definition 2.1. We call U = {Uk}∞k=1 a degree-regular flag if nk < nk+1 forall k. Equivalently, a flag is degree-regular if it admits a basis {pk}∞k=1 such thatdeg pk < deg pk+1 for all k.

ORTHOGONAL POLYNOMIALS AND FLAGS 53

Proposition 2.1. A polynomial flag is degree-regular if and only if the codi-mension sequence obeys �k ≤ �k+1 for all k.

Henceforth, we will always assume that all flags under discussion are degree-regular.

Definition 2.2. We say that the codimension of a polynomial flag is semi-stable if the codimension sequence �k admits an upper bound. If so, we call � =limk �k the codimension of the flag. We say that codimension is stable if �k = � isconstant for all k.

Likewise, whenever we speak of the codimension of a polynomial flag, we alwaysassume that the flag has a semi-stable codimension.

Let D2(U) denote the vector space of 2nd order differential operators withrational real-valued coefficients

(3) T (y) = p(x)y′′ + q(x)y′ + r(x)y, p, q, r ∈ R(x)

that preserve U ; i.e., T (Uk) ⊂ Uk for all k. If p, q, r are polynomials, we call T apolynomial operator. Equivalently, an operator is non-polynomial if and only if ithas a pole.

Definition 2.3. We say that U is imprimitive if it admits a common factor;i.e., U is spanned by {qpk} where q(x) is a polynomial of degree ≥ 1. Otherwise,we call U primitive.

At this juncture, it is important to state the following two Propositions.

Proposition 2.2. Let T (y) = py′′ + qy′ + ry be a differential operator suchthat

T (yi) = gi, i = 1, 2, 3,

where yi, gi are polynomials with y1, y2, y3 linearly independent. Then, p, q, r arerational functions with the Wronskian

W (y1, y2, y3) = det

⎛⎝y′′1 y′1 y1y′′2 y′2 y2y′′3 y′3 y3

⎞⎠in the denominator.

Proof. It suffices to apply Cramer’s rule to solve the linear system⎛⎝y′′1 y′1 y1y′′2 y′2 y2y′′3 y′3 y3

⎞⎠⎛⎝pqr

⎞⎠ =

⎛⎝g1g2g3

⎞⎠�

We should also note that there is a natural linear isomorphism between the spaceD2(U) of a flag U spanned by {pk}∞k=1 and the space D2(U) of the imprimitive flag

U spanned by {qpk}∞k=1.

Proposition 2.3. Let T (y) be an operator that preserves U . Then, the gauge-

equivalent operator T = qTq−1 preserves U .Proposition 2.2 makes clear why we restrict our definition of D2 to operators

with rational coefficients. Proposition 2.3 explains the need for primitive flags;these are the canonical representatives for the equivalence relation under gaugetransformations.


Definition 2.4. Let U be a polynomial flag of semi-stable codimension. Wesay that U is an exceptional flag if U is primitive and if D2(U) does not preserve apolynomial flag of smaller codimension.

Here are some examples to illustrate the above definitions.

Example 2.1. The codimension 1 flag spanned by 1, x2, x3, . . . , is exceptionalbecause the non-polynomial operator

T (y) = y′′ − 2y′/x

preserves the flag. Since �1 = 0, �k = 1, k ≥ 2, the codimension of this flag is notstable, but only semi-stable.

Example 2.2. By contrast, the flag spanned by x + 1, x2, x3, . . . has a stablecodimension � = 1. This flag is exceptional because it is preserved by the non-polynomial operator

T (y) = y′′ − 2(1 + 1/x)y′ + (2/x)y.

Example 2.3. Let hk(x) denote the degree k Hermite polynomial. The codi-mension 1 flag spanned by h1, h2, h3, . . . is not exceptional. The flag is preservedby the operator T (y) = y′′ − xy′, but this operator also preserves the standard,codimension zero, polynomial flag.

Example 2.4. The codimension 1 flag spanned by x, x2, x3, . . . is not excep-tional because x is a common factor. The flag is preserved by the operator

T (y) = y′′ − 2y′/x+ 2y/x2.

However, observe that T = xTx−1 where T (y) = y′′ is an operator that preservesthe standard, codimension zero, polynomial flag.

Example 2.5. Let

(4) y2k−1 = x2k−1 − (2k − 1)x, y2k = x2k − kx2, k = 2, 3, 4, . . .

Consider the codimension-2 flag spanned by 1, y3, y4, y5, y6, . . .. The degree se-quence of the flag is 0, 3, 4, 5, . . .; therefore the codimension is not stable, but merelysemi-stable. The flag is preserved by the following operators [17]:

T3(y) = (x2 − 1)y′′ − 2xy′,(5)

T2(y) = xy′′ − 2(1 + 2/(x2 − 1))y′,(6)

T1(y) = y′′ + x(1− 4/(x2 − 1))y′.(7)

The flag is exceptional, because T1 and T2 do not preserve the standard, codimen-sion zero flag. These operators cannot preserve a codimension 1 flag, because, aswill be shown in Lemma 3.2, an operator that preserves a codimension 1 flag canhave at most 1 pole.

2.2. Orthogonal polynomials.

Definition 2.5. We will say that a 2nd order operator T (y) is exactly solvableby polynomials if the eigenvalue equation

(8) T (y) = λy.

has infinitely many eigenpolynomial solutions y = yj with

deg yj < deg yj+1, j = 1, 2, . . . .


Let I = (x1, x2) be an open interval (bounded, unbounded, or semi-bounded)and let W (x)dx be a positive measure on I with finite moments of all orders. Wesay that a sequence of real polynomials {yj}∞j=1 forms an orthogonal polynomialsystem (OPS for short) if the polynomials constitute an orthogonal basis of theHilbert space L2(I,Wdx). If the codimension of the corresponding flag is stable,or semi-stable, we will say that the OPS has codimension m.

The following definition encapsulates the notion of a system of orthogonal poly-nomials defined by a second-order differential equation. Consider the boundaryvalue problem

− (Py′)′ +Ry = λWy(9)

limx→x±

i

(Py′u− Pu′y)(x) = 0, i = 1, 2,(10)

where P (x),W (x) > 0 on the interval I = (x1, x2), and where u(x) is a fixed poly-nomial solution of (9). We speak of a polynomial Sturm-Liouville problem (PSLP)if the resulting spectral problem is self-adjoint, pure-point and if all eigenfunctionsare polynomial.

If the eigenpolynomials span the standard flag, then we recover the classicalorthogonal polynomials, the totality of which is covered by Bochner’s theorem. Ifthe solution flag has a codimension m > 0, Bochner’s theorem no longer appliesand we encounter a generalized class of polynomials; we name these exceptional, orXm polynomials.

Given a PSLP, the operator

T (y) = W−1(Py′)′ −W−1Ry

is exactly solvable by polynomials. Letting p(x), q(x), r(x) be the rational coeffi-cients of T (y) as in (3), we have

P (x) = exp

(∫ x

q/p dx

),(11)

W (x) = (P/p)(x),(12)

R(x) = −(rW )(x),(13)

Hence, for a PSLP, P (x), R(x),W (x) belong to the quasi-rational class [15], mean-ing that their logarithmic derivative is a rational function.

Conversely, given an operator T (y) exactly solvable by polynomials and aninterval I = (x1, x2) we formulate a PLSP (9) by employing (11)–(13) as definitions,and by adjoining the following assumptions:

(PSLP1) P (x),W (x) are continuous and positive on I(PSLP2) Wdx has finite moments:

∫IxnW (x)dx <∞, n ≥ 0

(PSLP3) limx→xiP (x)xn = 0, i = 1, 2, n ≥ 0

(PSLP4) the eigenpolynomials of T (y) are dense in L2(I,Wdx).

These definitions and assumptions (PSLP1), (PSLP2) imply Green’s formula:

(14)

∫ x2

x1

T (f)gWdx−∫ x1

x1

T (g)f Wdx = P (f ′g − fg′)∣∣∣x2

x1

By (PSLP3) if f(x), g(x) are polynomials, then the right-hand side is zero. If f andg are eigenpolynomials of T (y) with unequal eigenvalues, then necessarily, they areorthogonal in L2(I,Wdx). Finally, by (PSLP4) the eigenpolynomials of T (y) arecomplete in L2(I,Wdx), and hence satisfy the definition of an OPS.


3. Codimension 1 flags

The key result in this paper is the following. An analogous theorem for poly-nomial subspaces, rather than flags, was proved in [20].

Theorem 3.1. Up to affine transformations, the flag spanned by {x+1, x2, x3, . . .}is the unique stable codimension 1 exceptional flag.

The proof proceeds by means of two lemmas.

Lemma 3.1. A primitive, codimension 1 polynomial flag is exceptional if andonly if D2 includes a non-polynomial operator.

Proof. It’s clear that a non-polynomial operator cannot preserve the standardpolynomial flag. The standard flag is the unique codimension zero flag. Therefore,if a non-polynomial operator preserves a primitive, codimension 1 flag, that flagmust be exceptional.

Let us prove the converse. Let T (y) be a polynomial operator that preserves apolynomial flag. Consider the degree homogeneous decomposition

(15) T (y) =

N∑d=−2

Td(y)

where

Td(y) = xd(αdx2y′′ + βdxy

′ + γdy)(16)

and where TN is non-zero. Since the operator is polynomial, we must have β−2 =γ−2 = γ−1 = 0. Also note that

(17) Td(xj) = (j(j − 1)αd + βdj + γd)x

j+d.

Let

yk(x) = xnk + lower deg. terms, k = 1, 2, . . .

be a basis of the flag. If N > 0, then the leading term TN (y) raises degree, andhence

TN (xnk) = 0

for all k. By (17), TN (y) annihilates at most 2 distinct monomials, a contradiction.Therefore, N ≤ 0, and we conclude that T (y) is a Bochner-type operator (1).However, such an operator preserves the standard polynomial flag. Therefore, if Uis exceptional, then there is at least one operator in D2(U) that doesn’t preserve thestandard flag. By the above argument, this operator must be non-polynomial. �

Lemma 3.2. Let T (y) be a non-polynomial operator that leaves invariant acodimension 1 polynomial subspace U ⊂ Pn. If n ≥ 5, then T (y) has exactly onepole. Furthermore, up to a translation in x, a basis of U has one of the followingthree forms:

x, x2, x3, . . . , xn(18)

1 + ax, x2, x3, . . . , xn(19)

1 + a1x2, x+ a2x

2, x3, . . . , xn(20)


Proof. By applying a translation if necessary, there is no loss of generality inassuming that x = 0 is a pole of T (y). Since the codimension is 1, the subspaceadmits an order-reduced basis of the form

yj = xj−1 + aj−1xν , j = 1, . . . , ν

yj = xj , j = ν + 1 . . . n,

where 0 ≤ ν ≤ n. The matrix representation of this basis is a n × n matrix inrow-reduced echelon form. This matrix has n pivots and 1 gap in position ν.

Our first claim is that ν ≤ 2. Suppose not. Then

W (y1, y2, y3) = W (1, x, x2) + higher degree terms = 2 +O(x).

By assumption, T (yi) is a polynomial. Hence, Proposition 2.2 implies that x = 0is not a pole of the operator, a contradiction.

Having established that ν ≤ 2, we observe that x3, x4, x5 ∈ U . The Wronskianof these monomials is a multiple of x9. Therefore, by Proposition 2.2, x = 0 is theunique pole. �

We are now ready to give the proof of Theorem 3.1. Let U be an exceptionalpolynomial flag with stable codimension 1. By Lemma 3.1, there exists a non-polynomial operator T (y) that preserves the flag. By Lemma 3.2, this operator hasa unique pole. Without loss of generality we assume that x = 0 is the unique pole.We rule out possibility (18), because if this holds for even one Uk, k ≥ 5, then itmust hold for all k. This would imply that x, x2, x3, . . . is a basis of the flag — aviolation of the primitivity assumption.

Let us rule out possibility (20). Let T (y) be a non-polynomial operator thatpreserves the flag. Since x = 0 is the unique pole, we can decompose the operatorinto degree-homogeneous terms

T (y) =

N2∑d=N1

Td(y)

where Td(y) has the form shown in (16), and where TN1and TN2

are non-zero. Bythe argument used in the proof of Lemma 3.1, we must haveN2 ≤ 0. By assumption,T (1 + a1x

2), T (x + a2x2), T (x3) are polynomials. However, if N1 ≤ −4, then this

condition requires that

TN1(1) = TN1

(x) = TN1(x3) = 0,

which means that TN1is zero — a contradiction. If N1 = −3, then T−3 an-

nihilates 1, x, x5, a contradiction. If N1 = −2, then T−2 annihilates 1, x, x4,another contradiction. Similarly, if N1 = −1, then T−1 annihilates 1. HenceT−1(y) = α−1xy

′′ + β−1y′. This means that there is no pole — a contradiction.

Therefore, N1 = N2 = 0, but that means that T (y) is a polynomial operator —again, a contradiction. This rules out possibility (20).

This leaves (19) as the only possibility. Since we assume that the codimensionis stable, deg(1+ax) = 1 and hence a �= 0. We scale x to transform 1+ax to x+1.This concludes the proof of the main theorem.


3.1. X1 polynomials. The above theorem explains the origin of the adjective“exceptional” and leads directly to a more general class of orthogonal polynomials— outside the class described by Bochner’s theorem. Two families of orthogonalpolynomials arise naturally when we consider codimension 1 flags. We describethese X1 polynomials below.

In order to construct codimension 1 polynomial systems, we must consider D2

of the flag spanned by x+ 1, x2, x3, . . ..

Proposition 3.1. The most general 2nd order operator that preserves the flag{x+ 1, x2, x3, . . .} has the form

(21) T (y) = (k2x2 + k1x+ k0)y

′′ − (x+ 1)

(k1 +

2k0x

)y′ +

(k1 +

2k0x

)y,

where k2, k1, k0 are real constants.

See [19, Proposition 4.10] for the proof.Thus to obtain X1 orthogonal polynomials it suffices to determine all possible

values of k2, k1, k0 for which P (x),W (x) as given by (11) (12) satisfy the conditionsof a PSLP. This analysis is performed in [21]. In summary, non-singular weightsarise only for the case where k2x

2 + k1x+ k0 either has two distinct real roots or ifk2 = 0, k1 �= 0. The first case leads to the Jacobi X1 polynomials; the second leadsto the X1 Laguerre polynomials. In both cases, the polynomial flags span a densesubspace of the respective Hilbert space [21, Proposition 3.1, Proposition 3.3]. Wesummarize the key properties of these two families below.

3.2. X1-Jacobi polynomials. Let α �= β be real parameters such that

(22) α > −1, β > −1, sgnα = sgn β.

Set

(23) a =1

2(β − α), b =

β + α

β − α, c = b+ 1/a.

Note that, with the above assumptions, we have |b| > 1. Now let

(24) u1 = x− c, ui = (x− b)i, i ≥ 2.

Define the measure dμα,β = Wα,β dx where

(25) Wα,β =(1− x)α(1 + x)β

(x− b)2, x ∈ (−1, 1).

Since Wα,β > 0 for −1 < x < 1, the scalar product

(26) (f, g)α,β :=

∫ 1

−1

f(x)g(x) dμα,β,

is positive definite. Also note that the above measure has finite moments of allorders.

We now define the X1-Jacobi polynomials P(α,β)n , n = 1, 2, . . . as the sequence

obtained by orthogonalization of the flag spanned by u1, u2, u3, . . . with respect to


the scalar product (26), and by imposing the normalization condition1

(27) P (α,β)n (1) = n

(α+ n− 1

n

).

From their definition it is obvious that deg P(α,β)n = n. However, as opposed to the

ordinary Jacobi polynomials, the sequence starts with a degree one polynomial.Next, define the 2nd order operator

(28) Tα,β(y) = (x2 − 1)y′′ + 2a

(1− b x

b− x

)((x− c)y′ − y

),

An elementary calculation shows that this operator preserves the flag {ui} as definedabove. Indeed, the flag {ui} and the operator (28) are obtained from the flagx + 1, x2, x3, . . . and from the operator (21) via the following specialization andaffine transformation

(29) k2 = 1, k1 = −2ab, k0 = (1− b2)a, x→ −a(x− b).

Multiplying both sides of the equation

−Tα,β(y) = λy

by Wα,β leads to the following PSLP:

((1− x2)Wα,β y′)′ + 2a

(1− b x

b− x

)Wα,β y = λ Wα,β y,(30)

limx→1−

(1− x)α+1((x− c)y′ − y) = 0,(31)

limx→−1+

(1 + x)β+1((x− c)y′ − y) = 0.(32)

The boundary conditions select polynomial solutions. The self-adjoint form of (30)ensures that the solutions are orthogonal relative to dμα,β . Therefore, theX1 Jacobi

polynomials can also be described as polynomial solutions, y = P(α,β)n (x), of the

following 2nd order equation:

(33) Tα,β(y) = (n− 1)(α+ β + n)y.

3.3. X1-Laguerre polynomials. For k > 0, set

(34) v1 = x+ k + 1, vi = (x+ k)i, i ≥ 2

Define the measure dμk = Wk dx where

(35) Wk =e−xxk

(x+ k)2, x ∈ (0,∞)

and observe that Wk > 0 on the domain in question. Therefore, the following innerproduct is positive definite:

(36) (f, g)k :=

∫ ∞

0

f(x)g(x) dμk,

Also note that the above measure has finite moments of all orders.

1This convention differs from the one adopted in [21]. The change is made to conform withthe convention adopted below for the generalized Xm Jacobi polynomials.


We define the X1-Laguerre polynomials Lk,i, i = 1, 2, 3, . . . as the sequenceobtained by orthogonalization of the flag spanned by v1, v2, v3, . . . with respect tothe scalar product (36) and subject to the normalization condition

(37) Lk,n(x) =(−1)nxn

(n− 1)!+ lower order terms n ≥ 1.

Again, since we are orthogonalizing a non-standard flag, the X1-Laguerre polyno-mial sequence starts with a polynomial of degree 1, rather than a polynomial ofdegree 0.

Define the operator

(38) Tk(y) = −xy′′ +(x− k

x+ k

)((x+ k + 1)y′ − y

)and note that this operator leaves invariant the flag spanned by the vi. Indeedthe flag {vi} and (38) are obtained from the flag x + 1, x2, . . . and (21) via thespecializations and an affine transformation shown below:

(39) k2 = 0, k1 = −1, k0 = k, x → x+ k

The corresponding PSLP takes the form

(xWky′)′ +

(x− k

x+ k

)Wky = λy,(40)

limx→0+

xk+1((x+ k + 1)y′ − y) = 0,(41)

limx→∞

e−x((x+ k + 1)y′ − y) = 0.(42)

As before, the boundary conditions select polynomial solutions, while the self-adjoint form of (40) ensures that these solutions are orthogonal relative to dμk.Therefore, the X1 Laguerre polynomials can also be defined as polynomial solu-tions, y = Lk,n, of the following 2nd order equation:

(43) Tk(y) = (n− 1)y.

Having introduced the X1 Jacobi and Laguerre polynomials, we are able to statethe following corollary of Theorem 3.1. The proof is found in [21].

Theorem 3.2. The X1-Jacobi polynomials and the X1 Laguerre polynomialsare the unique orthogonal polynomial systems defined by a stable codimension-1PSLP.

4. Higher codimension flags

Even though the first examples of exceptional orthogonal polynomials involvecodimension-1 flags, recently announced examples [35] are proof that exceptionalorthogonal polynomials can span flags of arbitrarily high codimension. Anotherimportant development is the recent proof [30] that the codimension-1 familiescan be related to the classical orthogonal polynomials by means of a Darbouxtransformation. In a follow-up publication [22], it was shown that the new highercodimension examples are systematically derivable by means of algebraic Darbouxtransformations [17].

A closely related development involves the notion of shape-invariance [13], amethodology related to the study of exactly solvable potentials. The close con-nection between solvable potentials and orthogonal polynomials is well recognized;


consider the relationship between the harmonic oscillator and Hermite polynomials,for example.

In the orthogonal polynomial context, we have to factorize general second or-der operators, not just Schrodinger operators. The shape-invariance property ofthe classical potentials manifests as the Rodrigues’ formula for the correspondingpolynomials. It has been pointed out that all the potentials related to exceptionalorthogonal polynomials exhibit the shape-invariance property[30, 38], and thattherefore, like their classical counterparts, exceptional orthogonal polynomials havea Rodrigues’ formula. This phenomenon was studied and explained in [22], where itwas shown that the shape-invariance property of the Xm (exceptional, codimensionm) polynomials follows from the permutability property of higher-order Darbouxtransformations.

4.1. The Darboux transformation. In the remainder of this section, wereview the some key definitions and results related to Darboux transformations andshape-invariance, and then illustrate these ideas with the example of Xm Jacobipolynomials[39].

Consider the differential operators:

T (y) = py′′ + qy′ + ry(44)

A(y) = b(y′ − wy), B(y) = b(y′ − wy),(45)

where p, q, r, b, w, b, w are rational functions.

Definition 4.1. We speak of a rational factorization if there exists a constantλ0 such that

(46) T = BA+ λ0

If the above equation holds, we call

(47) T = AB + λ0.

the partner operator. We call

(48) φ(x) = exp

∫ x

w dx, w = φ′/φ

a quasi-rational factorization eigenfunction and b(x) the factorization gauge.

The reason for the above terminology is as follows. By (46),

(49) T (φ) = λ0φ;

hence the term factorization eigenfunction. Next, consider two factorization gaugesb1(x), b2(x) and let T1(y), T2(y) be the corresponding partner operators. Then,

T2 = μ−1T1μ, where μ(x) = b1(x)/b2(x).

Therefore, the choice of b(x) determines the gauge of the partner operator. This iswhy we refer to b(x) as the factorization gauge. Also, note that in [17] the aboveconstruction was referred to as an algebraic Darboux transformation. However, inlight of the recently recognized role played by operators with rational coefficients,the term rational factorization seems to be preferable.


Proposition 4.1. Let T (y) be a 2nd order operator exactly solvable by polyno-mials, and let φ(x) be a quasi-rational factorization eigenfunction with eigenvalueλ0. Then, there exists a rational factorization (46) such that the partner operatoris also exactly solvable by polynomials, and such that the partner flag is primitive(no common factors).

Proof. Let w(x) = φ′(x)/φ(x) and let b(x) be an as yet unspecified rationalfunction. Set

w = −w − q/p+ b′/b,(50)

b = p/b,(51)

and let A(y), B(y) be as shown in in (45). An elementary calculation shows that(46) holds. Let y1, y2, . . . be a degree-regular basis of the eigenpolynomials of T (y).We require that the flag spanned by A(yj) be polynomial and primitive (no commonfactors). Observe that if we take b(x) to be the reduced denominator of w(x), thenA(yj) is a polynomial for all j. However, this does not guarantee that A(yj) is freeof a common factor. That is a stronger condition, one that fixes b(x) up to a choiceof scalar multiple. Finally, the intertwining relation

(52) TA = AT

implies that the A(yj) are eigenpolynomials of the partner T . �Finally, let us derive the formula for the partner weight function.

Proposition 4.2. Suppose that a PSLP operators T (y) = py′′ + qy′ + ry is

related to a PSLP operator T (y) = py′′ + qy′ + ry by a rational factorization withfactorization eigenfunction φ(x) and factorization gauge b(x), Then the dual fac-torization gauge, factorization eigenfunction and weight function are given by

bb = p(53)

W/b = W/b,(54)

bφ = 1/(Wφ)(55)

Proof. Equation (53) follows immediately from (45) (46). Writing

(56) T (y) = py′′ + qy′ + ry,

equation (47) implies that

(57) w + w = −q/p+ b′/b = −q/p+ b′/b.

Hence,

(58) q = q + p′ − 2pb′/b.

From here, (54) follows by equations (11) (12). Equation (55) follows from (48). �The adjoint relation between A and B allows us to compare the L2 norms of

the two families.

Proposition 4.3. Let T (y), T (y) be PSLP operators related by a rational fac-torization (46) (47). Let {yj} be the eigenpolynomials of T (y) and let yj = A(yj)be the corresponding partner eigenpolynomials. Then

(59)

∫ x2

x1

A(yj)2 Wdx = (λ0 − λj)

∫ x2

x1

y2j Wdx


where x1 < x2 are the end points of the Sturm-Liouville problem in question.

Proof. As a consequence of (54), A and −B are formally adjoint relative tothe respective measures:

(60)

∫ x2

x1

A(f)g Wdx+

∫ x2

x1

B(g)f Wdx = (P/b)fg∣∣∣x2

x1

,

where P (x) is defined by (11) and where b(x) is the factorization gauge. By assump-

tion both W (x), W (x) are positive on (x1, x2). By (54), W = P/b2. In particular,the numerator b(x) cannot have any zeroes in (x1, x2). As well, if either x1, x2 are

finite, we cannot have b(xi) = 0, because that would imply that W is not squareintegrable near x = xi. Hence |1/b(x)| is bounded from above on (x1, x2). There-fore, if f, g are polynomials then the right-hand side of (60) vanishes by (PSLP3).Therefore,

(61)

∫ x2

x1

A(yj)2 Wdx = −

∫ x2

x1

B(A(yj))yj Wdx = (λ0 − λj)

∫ x2

x1

y2j Wdx

�

4.2. Shape-invariance. Parallel to the L2 spectral theory [8, 40], rationalfactorizations of a solvable operator T (y) can be categorized as formally state-deleting, formally state-adding, or formally isospectral. The connection betweenthese formal, algebraic Darboux transformations and their L2 analogues is discussedin [22]. We speak of a formally state-deleting transformation if the factorizationeigenfunction φ(x) is the lowest degree eigenpolynomial of T (y). We speak of aformally state-adding transformation if the partner factorization eigenfunction

φ(x) = exp

∫ x

w dx,

with w defined by (50), is a polynomial. We speak of a formally isospectral trans-

formation if neither φ nor φ are polynomials. Examples of all three types of factor-izations will be given below.

State-adding and state-deleting factorizations are dual notions, in the sense thatif the factorization of T is state-deleting, then the factorization of T is state-adding,and vice versa.

Definition 4.2. Let κ ∈ K be a parameter index set and let

(62) Tκ(y) = p(x)y′′ + qκ(x)y′ + rκ(x)y, κ ∈ K,

be a family of operators that are exactly solvable by polynomials. If this family isclosed with respect to the state-deleting transformation, we speak of shape-invariantoperators.

To be more precise, let πκ(x) = yκ,1(x) be be the corresponding eigenpolyno-mial of lowest degree. Without loss of generality, we assume that the correspondingspectral value is zero, and let

(63) Tκ = BκAκ, Aκπκ = 0

be the corresponding factorization. Shape-invariance means that there exists aone-to-one map h : K → K and real constants λκ such that

(64) Th(κ) = AκBκ + λκ.


In accordance with (11), define

(65) Pκ(x) = exp

(∫ x

qκ/p

)Let bκ(x) denote the shape-invariant factorization gauge; i.e.;

(66) Aκ(y) = (bκ/πκ)W(πκ, y),

where

(67) W(f, g) = fg′ − f ′g

denotes the Wronskian operator. By equation (64),

(68) qh(κ) = qκ + p′ − 2pb′κ/bκ.

It follows that

(69) pPκ/Ph(κ) = b2κ.

Below, we will use this necessary condition to derive the factorization gauge of ashape-invariant factorization.

4.3. Jacobi polynomials. The operator

(70) Tα,β(y) := (1− x2)y′′ + (β − α+ (α+ β + 2)x)y′,

preserves the standard flag, and hence is exactly solvable by polynomials. The clas-

sical Jacobi polynomials P(α,β)n (x), α, β > −1, n = 0, 1, 2, . . . are the corresponding

eigenpolynomials:

(71) Tα,βP(α,β)n = −n(n+ α+ β + 1)P (α,β)

n ,

subject to the normalization

P (α,β)n (1) =

(n+ α

n

)The L2 orthogonality is relative to the measure Wα,β(x)dx where

(72) Wα,β(x) = (1− x)α(1 + x)β, x ∈ (−1, 1).The classical operators are shape-invariant by virtue of the following factoriza-

tions:

Tα,β = Bα,βAα,β,(73)

Tα+1,β+1 = Aα,βBα,β + α+ β + 2 where(74)

Bα,β(y) = (1− x2)y′ + (β − α+ (α+ β + 2)x)y,(75)

= (1− x)−α(1 + x)−β(y(1− x)α+1(1 + x)β+1

)′(76)

Aα,β(y) = y′.(77)

As a consequence, Bα,β(y) acts as a raising operator:

(78) Bα,βP(α+1,β+1)n = −2(n+ 1)P

(α,β)n+1 ,

and Aα,β(y) as a lowering operator:

(79) P (α,β)n

′ =1

2(1 + α+ β + n)Pα+1,β+1,n−1.


The classical Rodrigues’ formula, namely

(80) (1− x)−α(1− x)−β dn

dxn

((1− x)α+n(1− x)β+n

)= (−2)nn!P (α,β)

n (x),

follows by applying n iterations of the raising operator to the constant function.The quasi-rational solutions of Tα,β(y) = λy are known [4, Section 2.2]:

φ1(x) = P (α,β)m (x), λ0 = −m(1 + α+ β +m)

φ2(x) = (1− x)−α(1 + x)−βP−α,−β,m(x) λ0 = (1 +m)(α+ β −m)

φ3(x) = (1− x)−αP−α,β,m(x) λ0 = (1 + β +m)(α−m)

φ4(x) = (1− x)−βPα,−β,m(x), λ0 = (1 + α+m)(β −m)

where m = 0, 1, 2, . . .. The corresponding factorizations were analyzed in [16]. Ofthese, φ1 with m = 0 corresponds to a state-deleting transformation and under-lies the shape-invariance of the classical Laguerre operator and the correspondingRodrigues formula. For m > 0, the φ1 factorization eigenfunctions yield singularoperators and hence do not yield novel orthogonal polynomials. The φ2 familyresults in a state-adding transformation. The resulting flags are semi-stable, like inExample 2.5; see [17] for a discussion. The type φ3 φ4 factorizations result in novelorthogonal polynomials, provided α, β satisfy certain inequalities. These familieswere referred to as the J1, J2 Jacobi polynomials in [37]. The two families arerelated by the transformations α↔ β, x → −x. We therefore focus only on the φ3

factorization; no generality is lost.The derivations that follow depend in an elementary fashion on the following

well-known identities of the Jacobi polynomials. We will apply them below withoutfurther comment.

P(α,β)0 (x) = 1, P (α,β)

n (x) = 0, n ≤ −1,(81)

P (α,β)n (x) = (−1)nP (α,β)

n (−x),(82)

(x− 1)P (α,β)m

′(x) = (α+m)P (α−1,β+1)m (x)− αP (α,β)

m (x)(83)

P (α,β)n

′(x) =1

2(1 + α+ β + n)Pα+1,β+1,n−1(x),(84)

P (α,β−1)n (x)− P (α−1,β)

n (x) = P(α,β)n−1 (x)(85)

4.4. The Xm Jacobi polynomials. Fix an integer m ≥ 1 and α, β > −1,and set

(86) ξα,β,m = P (−α,β)m (x)

Take φ3(x) as the factorization eigenfunction and take

b(x) = (1− x)ξα,β,m

as the factorization gauge. Applying (45) (50) and the identities (81)-(85) , weobtain the following rational factorization of the Jacobi operator (28):

Tα,β = Bα,β,mAα,β,m − (m− α)(m+ β + 1) where(87)

Aα,β,m(y) = (1− x)ξα,β,m y′ + (m− α)ξα+1,β+1,m y(88)

Bα,β,m(y) = ((1 + x)y′ + (1 + β)y)/ξα,β,m(89)


By (55), φ(x) = (1 + x)−1−β is the dual factorization eigenfunction. Since nei-

ther φ3(x) nor φ(x) is a polynomial, (87) is an example of a formally isospectralfactorization. The corresponding partner operator is shown below:

Tα,β,m = Aα+1,β−1,mBα+1,β−1,m − (m− α− 1)(m+ β),(90)

Tα,β,m(y) = Tα,β(y)− 2ρα+1,β−1,m((1− x2)y′ + b(1− x) y)(91)

+m(a− b−m+ 1)y, where

ρα,β,m = ξ′α,β,m/ξα,β,m(92)

=1

2(1− α+ β +m) ξα−1,β+1,m−1/ξα,β,m(93)

Based on the above factorization, we define the Xm, exceptional Jacobi poly-nomials to be

P (α,β,m)n =

(−1)m+1

α+ 1 + jAα+1,β−1,mP

(α+1,β−1)j , j = n−m ≥ 0(94)

= (−1)m[1 + α+ β + j

2(α+ 1 + j)(x− 1)P (−α−1,β−1)

m P(α+2,β)j−1(95)

+1 + α−m

α+ 1 + jP (−2−α,β)m P

(α+1,β−1)j

](96)

By construction, these polynomials satisfy

(97) Tα,β,mP (α,β,m)n = −(n−m)(1 + α+ β + n−m)P (α,β,m)

n

With the above definition, the generalized Jacobi polynomials obey the followingnormalization condition

(98) P (α,β,m)n (1) =

(α+ n−m

n

)(n

m

), n ≥ m.

Note that the Xm operators and polynomials extend the classic family:

Tα,β,0(y) = Tα,β(y)(99)

P (α,β,0)n = P (α,β)

n .(100)

The L2 norms of the classical polynomials are given by∫ 1

−1

[P (α,β)n (x)

]2(1− x)α(1 + x)βdx = Nα,β

n

where

Nα,βn =

2α+β+1Γ(α+ 1 + n)Γ(β + 1 + n)

n!(α+ β + 2n+ 1)Γ(α+ β + n+ 1)

By (54), the weight for the Xm Jacobi polynomials is given by

(101) Wα,β,m(x) =(1− x)α(1 + x)β

ξα+1,β−1,m(x)2

In order for L2 orthogonality to hold for the generalized polynomials, we restrict α, βso that the denominator in the above weight is non-zero for −1 < x < 1. We alsowant to avoid the degenerate cases where x = ±1 is a root of ξα+1,β−1,m. To ensurethat we obtain a codimension m flag, we also demand that deg ξα+1,β−1,m = m.


The proof of the following Proposition follows from the analysis in [41, Chapter6.72].

Proposition 4.4. Suppose that α, β > −1. Then deg ξα+1,β−1,m = m andξα+1,β−1,m(±1) �= 0 if and only if β �= 0 and

(102) α, α− β −m+ 1 /∈ {0, 1, . . . ,m− 1}

Proposition 4.5. Suppose that the conditions of the preceding Propositionhold. The polynomial ξα+1,β−1,m(x) has no zeros in (−1, 1) if and only if α > m−2and

sgn(α−m+ 1) = sgn(β).

This is a good place to compare the above results to the parameter inequalitiesimposed in [37, 39]. These references impose the condition

α > β > m− 1/2.

Unlike Proposition 4.5, this condition fails to describe the most general non-singularweight Wα,β,m. Consider the following examples:

W1/3,−1/2,2 = 2882(1− x)1/3(1 + x)−1/2

(7x2 + 2x− 41)2(103)

W5/4,1/2,2 = 1282(1− x)5/4(1 + x)1/2

(5x2 − 14x+ 29)2(104)

Neither of the above examples satisfy the parameter inequalities of [37, 39], butboth weights are non-singular on (−1, 1) and have finite moments of all orders. Onthe other hand, the parameter values m = 2, α = 3/2, β = 1/2 give

W3/2,1/2,2(x) =3

8(1− x)3/2(1 + x)1/2,

P(3/2,1/2,2)2+k =

3

8P

(3/2,1/2)k , k ≥ 0.

In other words, for certain singular values of the parameters, the codimension isactually less than m, and in some instances (such as the one above) even yield theclassical polynomials. The condition (102) must be imposed in order to avoid suchsingular possibilities.

Proposition 4.6. Suppose that α, β > −1 satisfy the conditions of Proposi-tions 4.4 and 4.5 The L2 norms of the Xm Jacobi polynomials are given by∫ 1

−1

[P

(α,β,m)m+k (x)

]2Wα,β,mdx =

(1 + α+ k −m)(β +m+ k)

(α+ 1 + k)2Nα+1,β−1

k , k ≥ 0.

Proof. This follows directly from (59). �

We summarize the above findings as follows.

Theorem 4.1. Let m > 1 and α, β > −1 be such that α > m− 2, sgn(α−m+1) = sgn(β) and such that (102) holds. Let U be the stable, codimension m flag

spanned by polynomials y(x) such that (1+x)y′+βy is divisible by P(−α−1,β−1)m (x).

Let Wα,β,m(x) be the weight defined by (54). Then, the Xm Jacobi polynomials, asdefined by (94) are the orthogonal polynomials obtained by orthogonalizing the flag

U relative to the weight Wα,β,m(x) and subject to the normalization condition (98).


Finally, let us discuss shape-invariance of the generalized Jacobi operators. Thefollowing Proposition was proved in [22].

Proposition 4.7. Let A, B be the operators defined in (108) (109). Then

(105) Tα,β,m = Bα,β,mAα,β,m, Aα,β,mP (α,β,m)m = 0

is the state-deleting factorization of the Xm Jacobi operator. Furthermore,

(106) Tα+1,β+1,m = Aα,β,mBα,β,m + α+ β + 2,

is the dual state-adding factorization.

In essence, we are asserting that the generalized operators obey the same shape-invariance relations as their classical counterparts; c.f., equations (73) (74).

The proof of Proposition 4.7 relies on the permutability property of higherorder Darboux transformation and goes beyond the scope of this survey. We limitourselves to explicitly deriving the raising and lowering operators used in the abovefactorization.

We already know the factorization eigenfunction:

φ(x) = P (α,β,m)m (x) = (−1)m 1 + a−m

1 + aP (−α−2,β)m (x).

We make use of (69) to determine the factorization gauge. Making use of the factthat h(α, β) = (α+ 1, β + 1) we obtain

(107) b(x) =ξα+2,β,m

ξα+1,β−1,m.

We use Proposition 4.2 to derive the dual factorization gauge

b(x) = (1− x2)ξα+1,β−1,m

ξα+2,β,m

and the dual factorization eigenfunction,

φ(x) = (1− x)−α−1(1 + x)−β−1ξα+1,β−1,m.

We thereby obtain

Aα,β,m(y) =ξα+2,β,m

ξα+1,β−1,m(y′ − ρa+2,b,my)(108)

Bα,β,m(y) = (1− x2)ξα+1,β−1,m

ξα+2,β,m

[y′ −

(ρα+1,β−1,m +

α+ 1

1− x− β + 1

1 + x

)y

](109)

These shape-invariant factorizations serve as a good illustration of the duality be-tween formal state-adding and state-deleting transformations; here φ(x) is a poly-

nomial but φ(x) is merely a quasi-rational function.As well, the shape-invariant factorization illustrates that b(x), the factorization

gauge, is not necessarily a polynomial. Here,

w = −φ′/φ = ρα+2,β,m.

The denominator is ξα+2,β,m but the transformation

y → ξα+2,β,m(y′ − wy), y = P (α,β,m)n , n ≥ m

would produce an imprimitive flag. The common factor is ξα+1,β−1,m, and that iswhy the correct factorization gauge is the rational function shown in (107).


As a consequence of these shape-invariant factorizations, we have the followinglowering and raising identities for the Xm Jacobi operators; c.f., (78) (79)

Bα,β,mP(α+1,β+1,m)m+k = 2(1 + k)P

(α,β,m)m+k+1 , k ≥ 0;

Aα,β,mP(α,β,m)m+k =

1

2(1 + α+ β + k)P

(α+1,β+1,m)m+k−1 .

Acknowledgements

We thank Ferenc Tookos for useful comments and suggestions. The researchof DGU was supported in part by MICINN-FEDER grant MTM2009-06973 andCUR-DIUE grant 2009SGR859. The research of NK was supported in part byNSERC grant RGPIN 105490-2004. The research of RM was supported in part byNSERC grant RGPIN-228057-2004.

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Departamento de Fısica Teorica II, Universidad Complutense de Madrid, 28040

Madrid, Spain

Department of Mathematics and Statistics, McGill University Montreal, QC, H3A

2K6, Canada

Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H

3J5, Canada


On the Supersymmetric Spectra of two Planar IntegrableQuantum Systems

M. A. Gonzalez Leon, J. Mateos Guilarte, M.J. Senosiain,and M. de la Torre Mayado

“ Nil actum credens cum quid superesset agendum”“Nothing has been done if something remains to be done”

Abstract. Two planar supersymmetric quantum mechanical systems builtaround the quantum integrable Kepler/Coulomb and Euler/Coulomb prob-lems are analyzed in depth. The supersymmetric spectra of both systems areunveiled, profiting from symmetry operators not related to invariance withrespect to rotations. It is shown analytically how the first problem arisesat the limit of zero distance between the centers of the second problem. Itappears that the supersymmetric modified Euler/Coulomb problem is a quasi-isospectral deformation of the supersymmetric Kepler/Coulomb problem.

Contents

1. Introduction2. N = 2 supersymmetric planar quantum systems3. The planar quantum Kepler/Coulomb problem and supersymmetry4. The planar quantum Euler/Coulomb problem and supersymmetry5. Two center collapse in one center6. Further commentsAcknowledgementsReferences

1. Introduction

During the last forty years a very interesting jump from symmetry to super-symmetry has taken place, determining theoretic particle spectra in quantum fieldtheories with extremely appealing characteristics, see e.g. [1]. Unlike many quan-tum field theoretical models, the supersymmetric systems are frequently amenableto non-perturbative treatments, see e.g. [2], but the main feature is that fermionsand bosons are jointly assembled in multiplets, a fact, although suggestive, that has

2010 Mathematics Subject Classification. Primary 81Q60, 81Q80.Key words and phrases. Supersymmetric quantum mechanics, One Coulombian center of

force, Two Coulombian centers of force, Schrodinger equation separability, Razavy and Whittaker-Hill equations.


73

74 M.A.G. LEON, J.M. GUILARTE, M.J. SENOSIAIN, AND M. DE LA TORRE

not yet experimentally confirmed. Thus, mechanisms of spontaneous supersymme-try breaking must be investigated in the search for explanations of the apparent lackof supersymmetry in nature. In a series of papers, Witten, [3], [4], and [5], proposedthe analysis of this phenomenon in the simplest possible setting: supersymmetricquantum mechanics. A new area of research in quantum mechanics was born, withfar-reaching consequences both in mathematics and physics. The relation betweenthe Dirac operator in electromagnetic and/or gravitational fields - the supercharge- with the Klein-Gordon operator - the supersymmetric Hamiltonian - provided aguide for the building of supersymmetric quantum mechanical systems. The fac-torization method of identifying the spectra of Schrodinger operators by meansof first-order differential operators, see [6] for a review, is another antecedent ofsupersymmetric quantum mechanics that can also be traced back to the 19th cen-tury through the Darboux transform [7]. In its modern version, supersymmetricquantum mechanics prompted the study of many one-dimensional systems from aphysical point of view. A good deal of this work can be found in References [8], [9],[10], [11]. Several examples of this structure with emphasis in the semi-classicalbehavior of non-harmonic oscillators have been worked out in [12].

The formalism of physical supersymmetric systems with more than one boso-nic/fermionic pairs of degrees of freedom was first developed by Andrianov, Ioffe andcoworkers in a series of papers, [13], [14], published in the eighties. Factorability,even though essential in N-dimensional SUSY quantum mechanics, is not so effectiveas compared with the one-dimensional situation. Some degree of separability isalso necessary to achieve analytical results. For this reason we started a researchprogram in the two-dimensional supersymmetric classical mechanics of Liouvillesystems, [15]; i.e., those systems separable in elliptic, polar, parabolic, or Cartesiancoordinates, see papers [16] and [17]. We therefore follow this path in the quantumdomain for Type I Liouville models in [18].

Nevertheless, the authors from Saint Petersburg University mentioned aboveconsidered from the earlier eighties higher-than-one-dimensional SUSY quantummechanics from the point of view of the factorization of N-dimensional quantumsystems, [19], [20]. Ioffe et al. also studied the interplay between supersymme-try and integrability in quantum and classical settings in other types of model inReferences [21], [22], [23]. In these papers, a new structure was introduced [24]:second-order (and higher-order) supercharges provided intertwined scalar Hamil-tonians even in the two-dimensional (and higher-dimensional) case, see the reviewpapers [25] and [26]. This higher-order SUSY algebra allows for new forms ofnon-conventional separability in two dimensions. There are two possibilities: (1) asimilarity transformation performs the separation of variables in the superchargesand some eigen-functions (partial solvability) can be found, see [27], [28]. (2) Oneof the two intertwined Hamiltonian allows for exact separability: the spectrum ofthe other is known, [29], [30].

Our purpose in this paper is to describe planar supersymmetric systems - twobosonic/fermionic pairs of degrees of freedom - such that the Bose-Bose and Fermi-Fermi (scalar) Hamiltonians will be separable. In Reference [31] Eisenhart classifiedall the quantum systems with separable Schodinger equations in Cartesian, polar,parabolic, and elliptic coordinates. We shall address two planar supersymmetricseparable systems, one in polar, the other in elliptic coordinates.

ON THE SUSY SPECTRA OF TWO PLANAR INTEGRABLE QUANTUM SYSTEMS 75

We shall first consider the planar supersymmetric Kepler/Coulomb problemshowing the separability in polar coordinates. We strongly rely on the work byWipf et al in papers [32], [33] where they solve this problem in any dimension D byfinding a supersymmetric matrix Runge-Lenz vector and describing algebraicallythe spectral problem in terms of the irreducible representations of SO(D + 1). In-stead we shall attack the spectral problem in the Bose-Bose sector, finding thebound state energies in terms of the Casimir eigenvalues of the irreducible repre-sentations of SO(3), whereas the eigenfunctions are generalized Laguerre polyno-mials. The scattering eigenfunctions in this sector, as well as in the Fermi-Fermisector, are generic confluent hypergeometric functions given in terms of infinite se-ries. By acting with the supercharges, we provide in turn both the bound stateand scattering eigenfunctions in the Fermi-Bose Sector. We remark that, followinga previous work on the supersymmetric classical Kepler/Coulomb problem, [34],Heumann chose another superpotential [35] leading to a supersymmetric quantummechanical system where the Runge-Lenz vector is no longer an invariant even inthe Bose-Bose and Fermi-Fermi sectors.

In the second half of the paper we study a supersymmetric quantum mechan-ical system built from the classical Euler problem: a light particle moving in thegravitational field created by two fixed Newtonian centers of force restricted to theplane of the centers [36]. Besides Euler, this system attracted the interest of in-vestigators of stature such as Lagrange, Jacobi, Liouville, Darboux and others, see[36] to read a brief history of the subject, on a double front: 1) because of thepotential applications in celestial mechanics, e.g., as an intermediate step in thethree-body problem. 2) Because the Euler problem was a playground where theideas of integrability, curvilinear coordinates, Hamilton-Jacobi separability, of suchimportance in classical dynamics, were tested. All this was imported by Pauli [37]to the quantum domain in his research on the spectrum of the H+

2 hydrogen ionmolecule. A Chapter of Pauling and Wilson’s book on Quantum mechanics [38] isdevoted to the developments in this quantum problem up to the mid thirties of thepast century.

We shall address a supersymmetric quantum mechanical system such that thescalar Hamiltonians in the Bose-Bose and Fermi-Fermi sectors are related to thequantum mechanical Euler/Pauli Hamiltonian. We are guided by the separabil-ity of the Schrodinger equation in elliptic coordinates: half the sum and half thedifference of the distance to the centers. This is the main property of the Euler-Pauli Hamiltonian allowing for its integrability. We choose our scalar Hamiltoniansfulfilling this property but supersymmetry requires the energy to be non-negative.We are forced to add a “classical” piece to the EP potential energy that pushesthe ground state energy to zero. All this is achieved by the choice of a super-potential inspired in the Ioffe/Wipf et al superpotential for the supersymmetricKepler/Coulomb problem (our superpotential tends to the ABI/KLPW superpo-tential when the two centers collapse). In Reference [39], however, we exploredother possibilities in comparison with this superpotential.

A double change of variables to one-half of arccosh and arccos of the ellip-tic variables transforms the separated spectral problem into systems of entangledRazavy [40] and Whittaker-Hill [41] (three-term Hill, Razavy trigonometric) equa-tions. The two equations in each system are in principle independent, but they


are entangled because their parameters are determined from the integration con-stants -the energy and the separation constant- that allow one to formulate theHamiltonian spectral problem in terms of two separated ODE’s. To the best of ourknowledge, this change of variables was formulated for the first time in [42]. Weshall profit from the fact that for certain values of the parameters either the Razavyor the Whittaker-Hill equation are algebraic quasi-exactly solvable systems1. Morespecifically, the bound states of our system arise from values of the energy and theseparation constant that lead to Razavy equations belonging to this class of alge-braic QES potentials, see [44]. The algebraic QES potentials have correspondingquantum Hamiltonians that are elements of the enveloping algebra of a finite dimen-sional Lie algebra (Lie SL(2,R) in many cases), admitting an invariant finite moduleof smooth functions as irreducible representations [45]. These potentials were firststudied by Turbiner [46] and then completely classified by Gonzalez-Lopez, Kam-ran, and Olver [47]-[48]. The associated (weakly) orthogonal polynomials wereanalyzed in full generality in [49] and all this machinery was applied to study theRazavy trigonometric potential in [50].

Unlike in the non supersymmetric case, see [43]-[51], the Whittaker-Hill equa-tions unfortunately are not QES for the values of the parameters for which theRazavy equations are QES in the supersymmetric case. Thus, we can only give thesolutions in the form of infinite series following the theory of Hill equations, see e.g.[52]-[53]. The case of two centers of the same strength is exceptional: instead ofWhittaker-Hill equations we encounter Mathieu equations. The solutions can begiven analytically in terms of Mathieu Cosine and Sine functions2 [54] and it canbe explicitly checked that at the limit where the two centers collapse the one-centerwave functions are recovered.

The organization of the paper is as follows: after this long Introduction in thesecond Section §.2 we settle down to the framework of N = 2 supersymmetricQuantum Mechanics for systems of two degrees of freedom. Because each degreeof freedom can be labeled either as Bosonic or Fermionic, we have 2D = 4 typesof state. The Clifford algebra of R4 describes this situation perfectly and helpsus to define the supercharges, the Hamiltonian structure, and the Hilbert spaceof states. Section §.3 is fully devoted to discussing the planar supersymmetricKepler/Coulomb problem. The Bose-Bose bound state eigenfunctions arise as ir-reducible representations of the dynamical SO(3) symmetry associated with theRunge-Lenz vector whereas Bose-Fermi strictly positive bound states are obtainedvia the action of the Q† operator on the BB bound states (except the zero mode).The scattering states are also describe to unveil the whole spectrum of the super-symmetric Kepler/Coulomb problem. A direct analysis of the spectrum of the 4×4matrix Hamiltonian and symmetry operators is also included. In Section §.4 weaddress the same program in the supersymmetric modified Euler/Pauli two-centersystem. The bound state wave functions come from polynomial × exponential so-lutions of (infinite) Razavy equations multiplied by power series solutions of relatedWhittaker-Hill equations. In the case of two centers of the same strength, the WH

1In the non supersymmetric problem Demkov in [43] has shown that there are exceptionalvalues of the energy and the separation constant for which both equations in the pair are QESproviding finite solutions.

2All the conventions on special functions throughout the paper will follow Reference [54].


equations are replaced by Mathieu equations. In Section §.5 we show how the Ke-pler/Coulomb spectrum reappears at the d = 0 limit of the two centers. Finally, weoffer a final Section with further comments on interesting generalizations of theseclassically3 integrable models.

2. N = 2 supersymmetric planar quantum systems

2.1. N = 2 two-dimensional SUSY quantum mechanics. In this type ofquantum mechanical systems there are two pairs of canonically conjugated Bosonicoperators - giving the position and momentum of the particle - that we choose incoordinate representation:

pk = −i� ∂

∂xk, xk = xk , [xk, pj ] = i�δkj .

There are also two pairs of Fermionic operators - of physical dimensions: [ψk] =

M− 12 - taking care of the Fermionic degrees of freedom of the system. The Fermi

operators satisfy anti-commutation relations of the form:

(2.1) {ψk, ψl} = 0 = {ψ†k, ψ

†l } , {ψk, ψ

†l } =

1

mδkl , k, l = 1, 2 ,

showing that one operator is canonically conjugated to its adjoint operator.

The Fermionic Fock space is built from the vacuum state: ψk|0〉 = 0 , ∀k = 1, 2,the two degrees of freedom being in Bosonic states because |0〉 is an eigenstate of

zero eigenvalue of the Fermi number operator N =∑2

k=1 ψ†kψk: N |0〉 = 0|0〉. The

creation operators acting on |0〉 bring the system into one-particle states ψ†k|0〉 =

|1k〉, where one of the two degrees of freedom becomes Fermionic: N |1k〉 = |1k〉.The two-particle state - the two degrees of freedom in Fermionic states N |1112〉 =2|1112〉- are then obtained in a dual way related by Fermi statistics: ψ†

2|11〉 =

|1112〉 = −ψ†1|12〉 = −|1211〉.

The ortho-normality relations

〈0|0〉 = 〈11|11〉 = 〈12|12〉 = 〈1112|1112〉 = 1

〈11|0〉 = 〈12|0〉 = 〈12|11〉 = 〈1112|0〉 = 〈1211|11〉 = 〈1112|12〉 = 0

allow us to write the more general state in this finite Fermionic Fock space F =F0 ⊕F1 ⊕F2 in the form:

|f〉 = f0|0〉+2∑

k=1

f1k|1k〉+ f2|1112〉 , f0, f1k, f2 ∈ C .

The supersymmetric space of states is the direct product of F with the Hilbert spaceL2(R2): SH = H⊗ F = L2(R2)⊗ C4 = SH0 ⊕ SH1 ⊕ SH2. The supersymmetricwave functions read:

|Ψ(x1, x2)〉 = f0(x1, x2)|0〉+2∑

k=1

f1k(x1, x2)|1k〉+ f2(x1, x2)|1112〉

f0(x1, x2), f1k(x1, x2), f2(x1, x2) ∈ L2(R2) .

3Here we use the word classical in a non physical sense, i.e., classical does not refer to a class(versus quantal) of physical phenomena.


The standard procedure for introducing supersymmetric dynamics in this setupruns as follows: One first defines the supercharges4,

Q = iψk

(�

∂

∂xk+

∂W

∂xk

)= e−

W� Q0e

W� , Q† = iψ†

k

(�

∂

∂xk− ∂W

∂xk

)Q0 = i�ψk

∂

∂xk, [Q] = M

12LT−1

where

W (x1, x2) : R2 → R , [W ] = ML2T−1

is a real function called the superpotential.The supercharges are thus nilpotent first-order differential operators: Q2

o = 0 = Q2, that move states between the differentFermionic sectors of the space of states:

SH0

Q†

�Q

SH1

Q†

�Q

SH2 .

The super-Hamiltonian is defined to be

H =1

2{Q, Q†} = 1

2

(QQ† + Q†Q

)which implies: [H, Q†] = [H, Q] = 0. Therefore, the transformations generated by

Q and Q† are symmetries - called supersymmetries - of the dynamical system andthe supercharges are themselves constants of motion. Because [H, N ] = 0 there is

a Z2-grading of the dynamics given by the Klein operator F = (−1)N :

F |0〉 = |0〉 , F |1112〉 = |1112〉 , F |11〉 = −|11〉 , F |12〉 = −|12〉 .

In the classification of Supersymmetric Quantum Mechanics given by Kibler et al.in [55] our formalism ranks in the class of a complex super-charge, Q, with an

involution operator F 2 = I. It is also shown in Reference [55] that it is equivalentto another supersymmetric system with two real supercharges: this is the reasonfor the N = 2 in the title.

2.2. Clifford algebra representation. In order to skip abstract ket/braDirac algebra we represent the Fermi operators by means of the generators of theClifford algebra of R4:

ψ1 =1

2√m

(γ1 + iγ3

), ψ2 =

1

2√m

(γ2 + iγ4

)

ψ1 =1√m

⎛⎜⎜⎝0 1 0 00 0 0 00 0 0 10 0 0 0

⎞⎟⎟⎠ , ψ2 =1√m

⎛⎜⎜⎝0 0 1 00 0 0 −10 0 0 00 0 0 0

⎞⎟⎟⎠ .

One can check that this is a minimal realization of the Fermionic anticommutationrules (2.1) and the Fermionic Fock space becomes the space of four-component

4By [O] we shall denote the physical dimensions of the observable O.


Euclidean spinors with basis:

|0〉 →

⎛⎜⎜⎝1000

⎞⎟⎟⎠ , |11〉 →

⎛⎜⎜⎝0100

⎞⎟⎟⎠ , |12〉 →

⎛⎜⎜⎝0010

⎞⎟⎟⎠ , |1112〉 →

⎛⎜⎜⎝0001

⎞⎟⎟⎠ .

The supercharges are 4× 4-matrices of differential operators

Q =i√m

⎛⎜⎜⎝0 D1 D2 00 0 0 −D2

0 0 0 D1

0 0 0 0

⎞⎟⎟⎠ , Q† =i√m

⎛⎜⎜⎝0 0 0 0D1 0 0 0D2 0 0 00 −D2 D1 0

⎞⎟⎟⎠where Dk = �∂k + ∂W

∂xkand Dk = �∂k − ∂W

∂xk. The super-Hamiltonian is also a

4× 4-matrix of differential operators

H = H0 ⊗ I4 − �2∑

k=1

2∑l=1

∂2W

∂xk∂xlψ†kψl =

⎛⎜⎜⎝H0 0 0 0

0 H111 H12

1 0

0 H211 H22

1 0

0 0 0 H2

⎞⎟⎟⎠with a block-diagonal structure inherited from the eigen-spaces of the Fermi numberoperator:

N = ψ†1ψ1 + ψ†

2ψ2 =1

m

⎛⎜⎜⎝0 0 0 00 1 0 00 0 1 00 0 0 2

⎞⎟⎟⎠ .

Thus, in the F = +1 eigen-sectors of SH the Hamiltonian act by means of thescalar ordinary Schrodinger operators:

H0 ≡ H∣∣∣SH0

=1

2m

(−�2 �+∂1W∂1W + ∂2W∂2W + ��W

)H2 ≡ H

∣∣∣SH2

=1

2m

(−�2 �+∂1W∂1W + ∂2W∂2W − ��W

).

In SH1, however, the super-Hamiltonian reduces to the 2 × 2-matrix Schrodingeroperator:

H1 ≡ H∣∣∣SH1

=

(H11

1 H121

H211 H22

1

)=

(H0 − �

m∂21W − �

m∂1∂2W

− �

m∂2∂1W H0 − �

m∂22W

).

We see that all the interactions come from the gradient and the second-order partialderivatives of the superpotential.

3. The planar quantum Kepler/Coulomb problem and supersymmetry

Our first goal in this survey is the development of this formalism encompassingthe Hamiltonian of the Kepler/Coulomb problem.


3.1. The quantum Kepler/Coulomb Hamiltonian. We recall that theKepler/Coulomb Hamiltonian describing the quantum dynamics of one-electronatoms is:

K = − �2

2m�−α

r, α > 0 , [α] = ML3T−2 .

We re-scale positions and momenta to new variables xk → 1mαxk, pk → mαpk

with dimensions of [xk] = M2L4T−2 and [pk] = M−1L−2T . By this token we seethat the parameters m (particle mass) and α2 (strength of the coupling) factor outin the new Hamiltonian

K → mα2K = mα2

(−�2

2�−1

r

), [K] = M−2L−4T 2

and their only physical role is to set the energy scale.It is well known that this problem is superintegrable : The angular momentum

-one scalar in the plane-

L = −i�(x1

∂

∂x2− x2

∂

∂x1

)and the Runge-Lenz vector -two components in the plane-

A1 =1

2

(p2L+ Lp2

)− x1

r, A2 = −1

2

(p1L+ Lp1

)− x2

r

[L, A1] = i�A2 , [L, A2] = −i�A1

both commute with K:

[K, L] = [K, A1] = [K, A2] = 0 .

We remark that in our variables the physical dimensions of these operators are[L] = ML2T−1, [A1] = [A2] = 1 and recall that they close the SO(3) Lie algebra in

the space of negative energy (bound states) eigen-functions of KψE = EψE , E < 0:

[A1, A2] = −2i�(p21 + p22

2− 1

r

)L = −2i�KL

M1 =1√−2E

A1 , M2 =1√−2E

A2 M3 = L , [Ma, Mb] = i�εabcMc .

Moreover, because the SO(3) Casimir operator is

C2 = M21 + M2

2 + M23 = − 1

2K

(A2

1 + A22

)+ L2

and

A21 + A2

2 = 2K

(L2 +

�2

4

)+ 1

the Hamiltonian is given in terms of C2

K = −1

2· 1

C2 + �2

4

such that the SO(3) symmetry is not Noetherian but a dynamical symmetry. Onefinds immediately the bound state eigenvalues

Ej = −2

�2· 1

(2j + 1)2, j ∈ 1

2⊗ N ,


which must be multiplied by mα2 to find the physical bound state energies. Thebound state eigenfunctions, the SO(3) irreducible representations, will be given inthe next subsection.

3.2. The supersymmetric Kepler/Coulomb Hamiltonian. The super-potential proposed by Ioffe et al in [20] and by Wipf et al (independently) in [32]to build the supersymmetric version of the Kepler/Coulomb problem is5:

W (x1, x2) = −2

�r = −2

�

√x21 + x2

2 ,∂W

∂xk= −2

�xk

r,

∂2W

∂xk∂xl= − 2

�r

(δkl −

xkxl

r2

).

We also re-scale the Fermi operators ψk → 1√mψk , [ψk] = 1 to define the Kepler-

Coulomb supercharge:

Q = i

(�ψk

∂

∂xk− 2

�h

), [Q] = M−1L−2T

where

g = −�2

∂W

∂xk· ψk =

xk

r· ψk , g2 = 0 ,

{g†, g

}= 1 ,

(g†)2

= 0

is a “hedgehog” projection of the spin variables over the R2-plane. Explicitly,

g =1

r

⎛⎜⎜⎝0 x1 x2 00 0 0 −x2

0 0 0 x1

0 0 0 0

⎞⎟⎟⎠ , g†g =1

r2

⎛⎜⎜⎝0 0 0 00 x2

1 x1x2 00 x1x2 x2

2 00 0 0 x2

1 + x22

⎞⎟⎟⎠ .

The supersymmetric Kepler/Coulomb Hamiltonian reads:

(3.1) H =

(−�2

2�+

2

�2

)I4 −

1

r· X , X = [g, g†] = I4 − 2N + 2g†g ,

whereas the scalar Schrodinger operators in the Bosonic sectors are:

H0 = −�2

2�+

2

�2− 1

r= K +

2

�2, H2 = −�2

2�+

2

�2+

1

r= K +

2

�2+

2

r.

Thus, H0 is exactly the Kepler/Coulomb Hamiltonian plus a constant needed to setto zero the energy of the Bosonic ground state (zero mode); recall that supersymme-

try forbids negative energy eigen-states. H2, however, is also (modulo a constant)the Kepler/Coulomb Hamiltonian for a particle of opposite electric charge, say apositron. The force is repulsive and there will only be scattering states.

The matrix Schrodinger operator - already given in [20] circa 1984 - acting in

the two-dimensional sub-space of the Fermionic Fock space such that N = 1 is:

H1 =

(−�

2

2 �+ 2�2 − x2

1−x22

r3 − 2x1x2

r3

− 2x1x2

r3 −�2

2 �+ 2�2 +

x21−x2

2

r3

).

5In [35] Heumann proposed another superpotential which spoiled the Runge-Lenz vectorconservation.


3.3. N = 0 bound state eigenfunctions of H. The N = 0 bound stateeigenfunctions of H are exactly the eigenfunctions of H0, which are the same as thebound state eigenfunctions of K with displaced eigenvalues:

H0ψ(0)j =

(2

�2− 1

2· 1

�2(j(j + 1) + 14 )

)ψ(0)j =

(2

�2− 1

2· 1

�2(j + 12 )

2

)ψ(0)j

j =n

2∈ 1

2⊗ N , ˆH0ψ

(0)j = mα2E

(0)j ψ

(0)j =

2mα2

�2

(1− 1

(2j + 1)2

)ψ(0)j .

Because of the dynamical SO(3) symmetry of H0, the bound state eigen-functions,which are degenerated in energy, form irreducible representations characterized bytwo integer or half-integer numbers, j and m, providing the eigenvalues of theCasimir operator and M3 in common eigen-kets:

C2|j;m〉 = �2j(j + 1)|j;m〉 M3|j;m〉 = �m|j;m〉 , m : −j,−j + 1, · · · , j − 1, j .

Using polar coordinates r = +√

x21 + x2

2, ϕ = arctanx2

x1in coordinate represen-

tation the eigen-wave functions are of the form: ψ(0)jm(r, ϕ) = 〈r;ϕ|j;m〉. It is

not difficult to identify the highest weight eigen-wave functions in each irreduciblerepresentation. Let M+ = M1 + iM2 be the up-stairs ladder operator

M+ = �(j +

1

2

)A+ = �3

(j +

1

2

)eiϕ

{i

∂2

∂r∂ϕ− 1

r

∂2

∂ϕ2− 1

2

∂

∂r− i

2r

∂

∂ϕ− 1

�2

}which annihilates the highest weight state:

M+ψ(0)jj (r, ϕ) = 0 , ψ

(0)jj (r, ϕ) = 〈r;ϕ|j; j〉 = fj(r)e

ijϕ .

This first-order ODE is easily integrated

f ′j(r) =

(j

r− 2r

�2(2j + 1)

)fj(r) ⇒ fj(r) = rjexp

{− 2r

�2(2j + 1)

}and the normalized highest wave functions are:

(3.2) ψ(0)jj (r, ϕ) = 2

√2

π· uj

�2√(2j + 1)3(2j)!

· eijϕ · e−u2 , u =

4r

�2(2j + 1).

The down-stairs ladder operator M− = M1 − iM2 in polar coordinates reads:

M− = �3(j +

1

2

)e−iϕ

{−i ∂2

∂r∂ϕ− 1

r

∂2

∂ϕ2− 1

2

∂

∂r+

i

2r

∂

∂ϕ− 1

�2

}.

From the Lie algebra we see that

ψ(0)jm ∝ M j−m

− ψ(0)jj , m = j,m = j − 1, · · · ,m = −j + 1,m = −j M2j+2

− ψ(0)jj = 0

and from the ansatz ψ(0)jm(r, φ) = N j−|m|r|m|Pj−|m|(r)e

− 2r�2(2j+1) eimϕ, where the

N j−|m| are normalization constants, the recurrence relations

Pj−|m|(r) =

[(2|m|+ 1)(|m|+ 1)− 2(j + |m|+ 1)r

(2j + 1)�2

]Pj−|m|−1(r) +

(2|m|+ 1)r

2P ′j−|m|−1(r)

follow. Therefore, the N = 0 bound state eigen-functions are:

(3.3) ψ(0)jm(r, ϕ) =

2

�2

√2(j + |m|)!

(2j + 1)3(j − |m|)!π · u|m|

(2|m|)! · 1F1[|m| − j, 2|m|+ 1, u]eimϕe−u2 .


In (3.3) we have the Kummer confluent hypergeometric functions 1F1 for thosevalues of a and b such that the series

1F1[a, b, z] =∞∑k=0

(a)k(b)k

· zk

k!, (a)k =

Γ(a+ 1)

Γ(a− k + 1), (b)k =

Γ(b+ 1)

Γ(b− k + 1)

truncate to generalized Laguerre polynomials:

L2|m||m|−j(u) =

(j + |m|j − |m|

)1F1[|m| − j, 2|m|+ 1, u] .

In sum, the N = 0 bound state eigen-functions of the planar supersymmetricKepler-Coulomb problem are organized as (degenerated in energy multiplets) ir-reducible representations of SO(3) in L2(R+× S1) rather than in L2(S2) (sphericalharmonics).

3.3.1. Ortho-normality and lower energy levels. It is easy to check that thefollowing ortho-normality relations hold:

Table 1. Lower energy multiplets.

Energy Eigen-function

E(0)0 = 0 ψ

(0)00 (r, ϕ) = 2

�2

√2π

e− 2r

�2

E(0)12

= 32�2

⎧⎪⎨⎪⎩ψ(0)12

12

(r, ϕ) = 1�3

√2π

r1/2 e− r

�2 ei12ϕ

ψ(0)12

−12

(r, ϕ) = − 1�3

√2π

r1/2 e− r

�2 e−i 12ϕ

E(0)1 = 16

9�2

⎧⎪⎪⎪⎨⎪⎪⎪⎩ψ(0)11 (r, ϕ) = 8

9�4√

3πr e

− 2r3�2 eiϕ

ψ(0)10 (r, ϕ) = 2

9�4

√23π

(3�2 − 4r

)e− 2r

3�2

ψ(0)1−1(r, ϕ) =

89�4

√3π

r e− 2r

3�2 e−iϕ

E(0)32

= 158�2

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ψ(0)32

32

(r, ϕ) = 14�5

√3π

r3/2 e− r

2�2 ei32ϕ

ψ(0)32

12

= − 14�5

√π

r1/2 (r − 2�2) e− r

2�2 ei12ϕ

ψ(0)32

−12

= 14�5

√π

r1/2 (r − 2�2) e− r

2�2 e−i 12ϕ

ψ(0)32

−32

= − 14�5

√3π

r3/2 e− r

2�2 e−i 32ϕ

E(0)2 = 48

25�2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ψ(0)22 (r, ϕ) = 16

125 �6√15π

r2 e− 2r

5�2 e2iϕ

ψ(0)21 (r, ϕ) = 8

125 �6√

15πr(15 �2 − 4 r

)e− 2r

5�2 eiϕ

ψ(0)20 (r, ϕ) = 2

125 �6

√25π

(25 �4 − 40 �2 r + 8 r2

)e− 2r

5�2

ψ(0)2−1(r, ϕ) =

8125�6

√15π

r(15 �2 − 4 r

)e− 2r

5�2 e−iϕ

ψ(0)2−2(r, ϕ) =

16125�6

√15π

r2 e− 2r

5�2 e−2iϕ

84 M.A.G. LEON, J.M. GUILARTE, M.J. SENOSIAIN, AND M. DE LA TORRE∫ 2π

0

dϕ

∫ ∞

0

rdr (ψ(0)j1m1

)∗(r, ϕ)ψ(0)j2m2

(r, ϕ) = δj1j2δm1m2|j1 − j2| = 0, 1, 2, 3, · · ·∫ 4π

0

dϕ

∫ ∞

0

rdr (ψ(0)j1m1

)∗(r, ϕ)ψ(0)j2m2

(r, ϕ) = 2δj1j2δm1m2, |j1 − j2| =

1

2,3

2,5

2, · · · .

Note that in the case of one integer j1 and one half-integer j2 pairing it isnecessary to integrate the ϕ variable over 4π because of the double-valued repre-sentation.

We offer two Tables 1 and 2 with the lower energy multiplets, their 2D (cross-sections) and 3D plots.

Table 2. 3D Plots and cross-sections of the probability densities:N = 0.

Probability � = 1 � = 1density

|ψ(0)00 (x1, x2)|2

|ψ(0)12±

12

(x1, x2)|2

|ψ(0)1±1(x1, x2)|2

|ψ(0)10 (x1, x2)|2


3.4. N = 1 bound state eigenfunctions of H. The supersymmetric partnereigen-states belonging to SH1 with the same energies are of the form:

E(1)j =

2

�2

(1− 1

(2j + 1)2

)∈ (0,

2

�2) , j > 0 ,

ψ(1)jm(x1, x2) = �Q†ψ

(0)jm(x1, x2) =

⎛⎜⎜⎜⎝0

ψ(1)(1)jm (r, ϕ)

ψ(1)(2)jm (r, ϕ)

0

⎞⎟⎟⎟⎠Therefore,

(ψ(1)j1m1

(r, ϕ))† ψ(1)j2m2

(r, ϕ) = (ψ(0)j1m1

(r, ϕ))† �2 QQ† ψ(0)j2m2

(r, ϕ)

= 2 �2 (ψ(0)j1m1

(r, ϕ))† H ψ(0)j2m2

(r, ϕ) = 2 �2E(0)j2

(ψ(0)j1m1

(r, ϕ))† ψ(0)j2m2

(r, ϕ)

because Q ψ(0)jm(r, ϕ) = 0 and H ψ

(0)jm(r, ϕ) = E

(0)j ψ

(0)jm(r, ϕ). Integration over R2

gives ∫ 2π

0

dϕ

∫ ∞

0

r dr (ψ(1)j1m1

(r, ϕ))† ψ(1)j2m2

(r, ϕ)

= 2 �2 E(0)j2

∫ 2π

0

dϕ

∫ ∞

0

r dr (ψ(0)j1m1

(r, ϕ))† ψ(0)j2m2

(r, ϕ)

= 2 �2 E(1)j2

δj1j2δm1m2.

The normalized eigenspinors ψ(1)jm(r, ϕ)→ 1√

2�E(1)j

ψ(1)jm(r, ϕ)

ψ(1)(1)jm (r, ϕ) = i

2

�2

√8(j + |m|)!

(2j + 1)3((2j + 1)2 − 1)(j − |m|)!πu|m|

(2|m|)!eimϕ e−

u2[

cosϕ

{−j + |m|2|m|+ 1

1F1 [−j + |m|+ 1, 2|m|+ 2, u] + j 1F1 [−j + |m|, 2|m|+ 1, u]

}+

m

ue−iϕ

1F1 [−j + |m|, 2|m|+ 1, u]]

ψ(1)(2)jm (r, ϕ) = i

2

�2

√8(j + |m|)!

(2j + 1)3((2j + 1)2 − 1)(j − |m|)!πu|m|

(2|m|)!eimϕ e−

u2[

sinϕ

{−j + |m|2|m|+ 1

1F1 [−j + |m|+ 1, 2|m|+ 2, u] + j 1F1 [−j + |m|, 2|m|+ 1, u]

}+i

m

ue−iϕ

1F1 [−j + |m|, 2|m|+ 1, u]]

satisfy the spectral condition H ψ(1)jm(r, ϕ) = E

(1)j ψ

(1)jm(r, ϕ) ≡ H1 ψ

(1)jm(r, ϕ) and

form an orthonormal basis in SH1.Specifically, these two-component wave functions are linear combinations of

two contiguous generalized Laguerre polynomials. The reason is that Q does notcommute with the generators of the SO(3) symmetry: [Ma, Q

†] �= 0, ∀a = 1, 2, 3.

Therefore, the Q† action does not respect the SO(3) irreducible representations.Nevertheless, the spinorial wave functions are characterized by the quantum num-bers j and m, although the degenerated multiplets do not form irreducible repre-sentations of SO(3).

We show next the lower spinorial probability densities: Table 3.


Table 3. 3D Plots and cross-sections of the probability densities:N = 1.

Probability � = 1 � = 1density

|ψ(1)10 (x1, x2)|2

|ψ(1)1±1(x1, x2)|2

r|ψ(1)12±

12

(x1, x2)|2

|ψ(1)32±

32

(x1, x2)|2

3.5. Scattering states and supersymmetric Hodge spectral decompo-sition. On positive energy eigen-functions of the Kepler-Coulomb Hamiltonian Kthe normalized components of the Runge-Lenz vector and the angular momentumclose the SO(2, 1) Lie algebra:

M1 = − i√2E

A1 , M2 = − i√2E

A2 , M3 = L

[M1, M2] = −i�M3 , [M3, M1] = i�M2 , [M2, M3] = i�M1 .

To search for the scattering wave functions in SH0, the eigenfunctions of H0(−�2

2�+

2

�2− 1

r

)ψ(0)

E(0)(x1, x2) = E(0)ψ(0)

E(0)(x1, x2) , E(0) >2

�2,


we profit from the fact that the spectral problem is separable into polar coordinates.

The ansatz ψ(0)

E(0)(r, ϕ) = f(0)

E(0)(r)eimϕ leads to the ordinary differential equation

d2f(0)

E(0)

dr2+

1

r

df(0)

E(0)

dr− m2

r2f(0)

E(0)(r) +2

�2

(1

r+ E(0) − 2

�2

)f(0)

E(0)(r) = 0 .

Now defining the non-dimensional variable u =2√

2| 2�2 −E(0)|�

r we find the Bosonicscattering solutions

(3.4) f(0)

E(0)(r) = N(E(0))e−iu2 u|m|

1F1

⎡⎣|m| − 1

2− i

�√2| 2

�2 − E(0)|, 1 + 2|m|, iu

⎤⎦in terms of Kummer confluent hypergeometric functions.

Simili modo, we identify the scattering wave functions in SH2, the eigenfunc-tions of H2:(

−�2

2�+

2

�2+

1

r

)ψ(2)

E(2)(x1, x2) = E(2)ψ(2)

E(2)(x1, x2) , E(2) >2

�2

d2f2E(2)

dr2+

1

r

df2E(2)

dr− m2

r2f2E(2)(r) +

2

�2

(−1

r+ E(2) − 2

�2

)f2E(2)(r) = 0

f2E(2)(r) = N(E(2))e−iu2 u|m|

1F1

⎡⎣|m| − 1

2+

i

�√2| 2

�2 − E(2)|, 1 + 2|m|, iu

⎤⎦ .

The potential being repulsive, there are no bound states in SH2.The supersymmetry algebra now allows us to identify all the solutions of the

supersymmetric spectral problem HψE = EψE from the eigenfunctions of H0 andH2 with non-zero eigenvalue. The key observation is that there are two kindsof non-zero (strictly positive energy) eigenfunctions ψ→

E = Q†ψE ∈ Q†SH and

ψ←E = QψE ∈ QSH because, if E > 0:

HQ†ψE = Q†HψE = EQ†ψE , HQψE = QHψE = EQψE .

The structure of the spectrum is as follows:

• Ground states.There is a unique ground state -that belongs to SH0 and hence

Bosonic- of zero energy E(0)0 = 0: ψ

(0)00 (x1, x2) ∈ KerH .

• There exist Q†-exact eigenstates of three types(1) Q†-exact - henceforth, living in Q†SH - bound state eigen-spinors

that belong to SH1:

ψ(1)

E(1)j

(x1, x2) = Q†ψ(0)jm(x1, x2)

(2) Q†-exact - henceforth, living in Q†SH - scattering eigen-spinors thatbelong to SH1:

E(1)− = E(0) >

2

�2, ψ

(1)

E(1)−

(x1, x2) = Q†ψ(0)

E(0)(x1, x2)


(3) Q†-exact - henceforth, living in Q†SH - scattering wave-functionsthat belong to SH2:

E(2) = E(1)+ >

2

�2, ψ

(2)

E(2)(x1, x2) = Q†ψ(1)

E(1)+

(x1, x2)

• There exist Q-exact eigenstates also of three types(1) Q-exact bound states -henceforth belonging to QSH- but living inSH0:

E(0)j = E

(1)j , j > 0 , ψ

(0)jm(x1, x2) = Qψ

(1)

E(1)j

(x1, x2) .

(2) Q-exact - henceforth, living in QSH - scattering wave-functions thatbelong to SH0:

E(0) = E(1)− >

2

�2, ψ

(0)

E(0)(x1, x2) = Qψ(1)

E(1)−

(x1, x2)

(3) Q-exact - henceforth, living in Q†SH - scattering eigen-spinors thatbelong to SH1:

E(1)+ = E(2) >

2

�2, ψ

(1)

E(1)+

(x1, x2) = Qψ(2)

E(2)(x1, x2)

Because the eigenfunctions form a total set in each sub-space we have the decom-position a la Hodge of the supersymmetric space of states:

SH = QSH⊕

Q†SH⊕

KerH .

Figure 1. The spectrum of the Kepler/Coulomb Hamiltonian(left panel). The spectrum of the supersymmetric Kepler/CoulombHamiltonian (right panel).


3.6. Spin-statistics structure of the supersymmetric Kepler/Coulombproblem. We have unveiled the spectrum of the planar supersymmetric Kepler/Coulomb problem solving the spectral problem of the two scalar Hamiltonians andusing the supersymmetry algebra to obtain the eigenfunctions of the matrix opera-tor. It is convenient, however, to look at the system as a whole, i.e., search directlyfor the spectrum of the 4× 4-matrix supersymmetric Hamiltonian operator.

With this goal in mind we define the “spin” operator

1

2S = −i�

2

(ψ†1ψ2 − ψ†

2ψ1

)= −i�

2

⎛⎜⎜⎝0 0 0 00 0 1 00 −1 0 00 0 0 0

⎞⎟⎟⎠ .

Clearly, [S, N ] = 0, such that 12 S and N share eigenstates fulfilling a quantum

mechanical spin-statistics theorem. The Bosonic eigenstates of N are zero spineigenstates of 1

2 S, whereas the Fermionic eigenstates of N are one-half spin eigen-

states of 12 S:

1

2S

⎛⎜⎜⎝ψ(0)

000

⎞⎟⎟⎠ =1

2S

⎛⎜⎜⎝000

ψ(2)

⎞⎟⎟⎠ = 0 ,1

2S

⎛⎜⎜⎝0

ψ(1)

±iψ(1)

0

⎞⎟⎟⎠ =�2

⎛⎜⎜⎝0

ψ(1)

±iψ(1)

0

⎞⎟⎟⎠ .

Note also that neither the orbital angular momentum L nor the spin angularmomentum S commute with H. The “total” angular momentum is the quantuminvariant associated to simultaneous rotations of the Bosonic xk and Fermionic ψk

coordinates6 :

J = L+ S = −i�(x1

∂

∂x2− x2

∂

∂x1

)− i�

(ψ†1ψ2 − ψ†

2ψ1

).

Therefore, besides the fact that [J , N ] = 0, one can use [J , g] = [J , X] = 0 to showthat:

[J , Q] = [J , Q†] = [J , H ] = 0 ,

where X is defined as in (3.1):

X = [g, g†] = I4 − 2N + 2g†g .

We have two Clifford supersymmetric operators commuting with each other: H andJ . The supersymmetric system as a whole is integrable. Now, the challenge is tofind more Clifford differential operators commuting with the Hamiltonian. In [32]the authors found the supersymmetric version of the Runge-Lenz vector operator-henceforth, the supersymmetric KLPW vector operator-:

W1 =1

2

(p2J + J p2

)− x1

r· X , W2 = −1

2

(p1J + J p1

)− x2

r· X .

One could guess the step from L to J and the need for the factor X is also nosurprise given its role in the supersymmetric Hamiltonian H. A long computationensures that the two components of this 4 × 4-matrix vector differential operatorwill indeed commute with the Hamiltonian and with the Fermi number operator:

[Wk, Q] = [Wk, Q†] = [Wk, H] = [Wk, N ] = 0 .

6Recall that [L, xk] = �εkjxj and [S, ψk] = �εkj ψj , ε12 = −ε21 = 1, ε11 = ε22 = 0 .


Some work is also necessary to check that

[J , W1] = i�W2 , [J , W2] = −i�W1 , [W1, W2] = −2i�(H − 2

�2

)J .

Therefore, defining

Ck =Wk√

2( 2�2 − H)

the SO(3) Lie algebra is now closed -in the sub-space of states of energy in the

range E ∈ (0, 2�2 )- by the Clifford operators C1, C2, C3 = J

[Ca, Cb] = −i�εabcCc, a, b, c = 1, 2, 3

and the Casimir operator is the 4 × 4-matrix differential operator: C2 =1�2

[C2

1 + C22 + C2

3

]. In [32] the authors were able to find:

H∣∣∣QH =

1

2QQ† =

2

�2

(1− (1− 2N)2

(1− 2N)2 + 4C2

),

H∣∣∣Q†H =

1

2Q†Q =

2

�2

(1− (3− 2N)2

(3− 2N)2 + 4C2

)

such that E(0)j = 2

�2

(1− 1

(2j+1)2

)= E

(1)j , the bound state energies paired through

supersymmetry in SH0 and SH1, reappear.

4. The planar quantum Euler/Coulomb problem and supersymmetry

Our second task is to build a supersymmetric quantum mechanical systeminspired in the Euler/Coulomb problem: a massive/charged particle that moves ona plane under the influence of two fixed Newtonian/Coulombian centers, see nextFigure.

Figure 2. Location of the two centers and distances to the particlefrom the centers.


4.1. The quantum Euler Hamiltonian. The quantum Euler Hamiltonianis:

I1 = − �2

2m�−α1

r1− α2

r2.

Here, (x±1 = ±d, x±

2 = 0) are the locations of the centers in the x1-axis, andα1 = α ≥ α2 = δα are the center strengths. With no loss of generality, we assumeδ ∈ (0, 1] affording hetero-nuclear one-electron diatomic molecular ions. Thus,the strengths depend on the atomic numbers Z1 and Z2 of the atoms and the δparameter is a positive rational number less than or equal to one:

α = e2Z1 , δ =Z2

Z1≤ 1, , Z1, Z2 ∈ N∗ , Z1 ≥ Z2 .

Finally, the distances of the particle to the two centers are: r1 =√(x1 − d)2 + x2

2,

r2 =√(x1 + d)2 + x2

2.Unlike in the Kepler/Coulomb problem there is a parameter with dimensions

of length in the system: the distance between the centers d. This allows us to usenon dimensional spatial coordinates:

x1 → d x1 x2 → d x2 r1 → d r1 = d√

(x1 − 1)2 + x22 r2 → d r2 = d

√(x1 + 1)2 + x2

2 .

Note that there is also a fundamental action√mdα built from the parameters of the

system that provides a non dimensional Planck constant: � = �√mdα

. Assembling all

this together, the linear momentum and Hamiltonian operators go to pi →√

mαd p

and I1 → αd I1, where the new non dimensional operators are:

pi = −i�∂

∂xi, I1 = − �2

2

(∂2

∂x21

+∂2

∂x22

)− 1

r1− δ

r2.

In this problem there is a non-obvious symmetry operator I2 → mdαI2 wherethe non dimensional operator reads [15]:

I2 =1

2

(L2 − p22

)+

1− x1

r1+

(1 + x1)δ

r2.

Just as the Runge-Lenz vector is quadratic in the momenta but unlike in the Ke-pler/Coulomb problem there are no more invariants in the Euler system which,accordingly, is only integrable. Explicitly,

I2 = − �2

2

((x2

1 − 1)∂2

∂x22

+ x22

∂2

∂x21

− 2x1x2∂2

∂x1∂x2− x1

∂

∂x1− x2

∂

∂x2

)+

1− x1

r1+

(1 + x1)δ

r2

and a little algebra shows that: [I1, I2] = I1I2 − I2I1 = 0.

4.2. Separability of the Schrodinger equation in elliptic coordinates.Because of the quadratic in the momenta symmetry operator, we expect that theSchrodinger equation will be separable in some coordinate system on the plane. Toskip the singularities in the centers one can cover the plane by two open charts:the first chart is the open set R+ × N in R2/{(−1, 0)} = R+ × S1 where N isthe open North hemisphere in S1. The second chart is the open set R+ × S inR2/{(1, 0)} = R+ × S1 where S is the open South hemisphere. Both charts mustbe glued at the abscissa axis x2 = 0. Totally adapted to this topological situation


are the elliptic coordinates; the half-sum and half-difference of the distances to thecenters of the particle:

u =1

2(r1 + r2) ∈ (1,+∞) , v =

1

2(r2 − r1) ∈ (−1, 1)

that parametrize a two dimensional infinite strip E2. The Cartesian coordinatesare obtained through the change

x1 = uv ∈ (−∞,+∞) , x2 = ±√(u2 − 1)(1− v2) ∈ (−∞,+∞)

which is a two-to-one map -one per chart- from R2 to E2 except at the x2-axis, whichis one-to-one mapped at the boundary of E2: ∂E2 = {(u = 1, v), (u, v = ±1)}.

The Euler Hamiltonian in elliptic coordinates is of the separable form

I1 =1

u2 − v2

(Hu + Hv

)Hu = − �2

2

((u2 − 1)

∂2

∂u2+ u

∂

∂u

)− (1+ δ)u Hv = − �2

2

((1− v2)

∂2

∂v2− v

∂

∂v

)− (1− δ)v

and the symmetry operator also separates:

I2 =1

u2 − v2

[(u2 − 1)Hv − (1− v2)Hu

].

The ansatz ψE(u, v) = ηE(u)ξE(v) converts the spectral problem

(4.1) I1ψE(u, v) = EψE(u, v)

into separable:

−�2(u2 − 1)d2ηE

du2(u)− �2u

dηE

du(u)−

[2(1 + δ)u+ 2u2E

]ηE(u) = IηE(u)(4.2)

−�2(1− v2)d2ξE

du2(v) + �2v

dξE

du(v) +

[−2(1− δ)v + 2v2E

]ξE(v) = −IξE(v) .(4.3)

The Schrodinger PDE equation (4.1) becomes the two coupled ODE’s (4.2)-(4.3)

where the separation constant I is the eigenvalue of the symmetry operator I =−2I1 − 2I2.

We could try to solve (4.2)-(4.3) directly but we still perform the followingchange of variables:

(4.4) x =1

2arccoshu ∈ [0,∞) , y =

1

2arccosv ∈ [0,

π

2] .

Equation (4.2) becomes the Razavy equation (4.5) [40], and (4.3) becomes theRazavy trigonometric (4.6) or Whittaker-Hill equation [41]:

− d2ηE(x)

dx2+ (ζ cosh 2x−M)

2ηE(x) = ληE(x)(4.5)

d2ξE(y)

dy2+ (β cos2y −N)2ξE(y) = μξE(y) .(4.6)

The parameters in the Razavy equations (4.5) and (4.6) are defined in terms of theenergy and the eigenvalue of the symmetry operator in the form:

ζ =2√−2E�

, M2 = −2(1 + δ)2

�2E, λ = M2 +

4I

�2

β = −2√−2E�

, N2 = −2(1− δ)2

�2E, μ = N2 +

4I

�2.


We stress the following subtle point: the Razavy equations are defined for fixedζ, M , and λ or for fixed β, N , and μ. We obtain, however, Razavy equations forparameters determined from E and I. Therefore, we address an infinite number ofentangled Razavy and Razavy trigonometric equations.

In Reference [44] it is shown that the Razavy and Whittaker-Hill equationsfor M and N positive integers are quasi-exactly solvable (QES) systems and allthe finite solutions -polynomials times fast decreasing exponentials- are found byalgebraic means. Our strategy will be to use this information in the search for thebound state eigenvalues and eigenfunctions of the Euler/Pauli Hamiltonian. Ourresults have been partially published in [39]. Thus, we shall not repeat the analysishere. Instead, we shall develop the program in the supersymmetric version of theEuler problem.

4.3. The supersymmetric modified Euler/Coulomb Hamiltonian. In[39] we gave arguments for selecting the following superpotential

(4.7) W (x1, x2) = −2

�(r1 + δr2)

in order to develop a supersymmetric quantum mechanical system from two fixedcenters containing a mild deformation of the Euler/Coulomb system in the N = 0sector. In fact, from the superpotential partial derivatives

∂W

∂x1= − 1

2�

(x1 − 1

r1(1− δ) +

x1 + 1

r2(1 + δ)

)∂W

∂x2= − 1

2�

(x2

r1(1− δ) +

x2

r2(1 + δ)

)∂2W

∂x21

= − 1

2�

{(1

r1− (x1 − 1)2

4r31

)(1− δ) +

(1

r2− (x1 + 1)2

4r32

)(1 + δ)

}∂2W

∂x22

= − 1

2�

{(1

r1− x2

2

4r31

)(1− δ) +

(1

r2− x2

2

4r22

)(1 + δ)

}∂2W

∂x1∂x2=

1

2�

(x2(x1 − 1

4r31(1− δ) +

x2(x1 + 1)

4r32

)=

∂2W

∂x2∂x1

we obtain first the supercharges. In turn, the scalar Hamiltonians,

H0 = − �2

2�+

2

�2

[1 + δ2 + δ

(r1r2

+r2r1− 4

r1r2

)]− 1

r1− δ

r2

H2 = − �2

2�+

2

�2

[1 + δ2 + δ

(r1r2

+r2r1− 4

r1r2

)]+

1

r1+

δ

r2,

and the matrix Hamiltonian:

H1 =

⎛⎜⎜⎝12

(H0 + H2

)− (x1−1)2−x2

2r31

− δ(x1+1)2−x2

2r32

−2

(x2(x1−1)

r31

+ δx2(x1+1)

r32

)−2

(x2(x1−1)

r31

+ δx2(x1+1)

r32

)12

(H0 + H2

)+

(x1−1)2−x22

r31

+ δ(x1+1)2−x2

2r32

⎞⎟⎟⎠ .

are derived. Now, the rationale for the choice of the superpotential (4.7) is clearer:at the limit where the two centers are superposed, d = 0 and r1 = r2, the super-potential, the supercharges and the superHamiltonian become those of the super-symmetric Kepler/Coulomb problem (with non-dimensional strength 1 + δ insteadof 1).

In this one-parametric deformation of the Kepler problem a very subtle co-nundrum arises. In the Kepler/Coulomb case, the non-supersymmetric K and the


N = 0 scalar H0 Hamiltonians only differ in the shift by a constant necessary topush the ground state energy to zero as required by supersymmetry. In the Eu-ler/Coulomb case, I1 and H0 differ in a non-constant potential such that in theKepler limit becomes the constant shift7:

VS(x1, x2) =2mα2

�2

[1 + δ2 + δ

(r1r2

+r2r1

− 4d2

r1r2

)], limd→0

VS(x1, x2) =2mα2(1 + δ)2

�2.

The role of this potential is to shift the negative bound state energies in a harm-less way: H0, like I1, still admits a symmetry operator that is quadratic in themomenta and the supersymmetric spectral problem in SH0 is still separable in el-liptic coordinates !! In other words, we choose the superpotential in such a waythat the separability in elliptic coordinates of H0 is preserved even though we mustadd a “classical” piece -important when � tends to 0 - to the Euler/Coulomb non-supersymmetric Hamiltonian.

In fact, H0 and the symmetry operator I2 in elliptic coordinates are still of theform

H0 =1

u2 − v2

(H(0)

u + H(0)v

), I

(0)2 =

1

u2 − v2

[(u2 − 1)H(0)

v − (1− v2)H(0)u

]where H

(0)u and H

(0)v are now 8:

H(0)u = − �2

2

((u2 − 1)

∂2

∂u2+ u

∂

∂u

)+

2

�2(1 + δ)2(u2 − 1)− (1 + δ)u(4.8)

H(0)v = − �2

2

((1− v2)

∂2

∂v2− v

∂

∂v

)+

2

�2(1− δ)2(1− v2)− (1− δ)v .(4.9)

The separation ansatz ψ(0)E (u, v) = η

(0)E (u)ξ

(0)E (v) plugged into the supersym-

metric spectral problem in the N = 0 Bosonic subspace

H0ψ(0)E (u, v) = Eψ

(0)E (u, v)

reduces the PDE Schrodinger equation to the system of separated ODE’s[−�2(u2 − 1)

d2

du2− �2u

d

du+

(4(1 + δ)2

�2(u2 − 1)− 2(1 + δ)u− 2Eu2

)]η(0)E (u) = Iη

(0)E (u)(4.10)[

−�2(1− v2)d2

dv2+ �2v

d

dv+

(4(1− δ)2

�2(1− v2)− 2(1− δ)v + 2Ev2

)]ξ(0)E (v) = −Iξ

(0)E (v)(4.11)

where the separation constant I is the eigenvalue of the symmetry operator I =

−2H0 − 2I(0)2 .

4.4. Bound states from entangled Razavy and Whittaker-Hill equa-tions. The change of coordinates (4.4) transforms (4.10) and (4.11) respectively inthe Razavy and Whittaker-Hill (three-term Hill) equations

− d2η(0)E (x)

dx2+ (ζ cosh 2x−M)

2η(0)E (x) = λ η

(0)E (x)(4.12)

d2ξ(0)E (y)

dy2+ (βcos2y −N)2 ξ

(0)E (y) = μ ξ

(0)E (y) ,(4.13)

7We temporarily come back to dimensional coordinates in order to see the limit.8Obviously, the same situation happens in the N = 2 sector. H2 and I22 are given in the

same way in terms of H2u and H2

v that differ from H(0)u and H

(0)v in the sign of the last terms.


where the parameters are now:

ζ =2

�

√4

�2(1 + δ)2 − 2E , M2 =

2(1 + δ)2

2(1 + δ)2 − �2E, λ = M2 +

4

�2

(I +

4(1 + δ)2

�2

)β = − 2

�

√4

�2(1− δ)2 − 2E , N2 =

2(1− δ)2

2(1− δ)2 − �2E, μ = N2 +

4

�2

(I +

4(1− δ)2

�2

).

4.5. Eigenfunctions from the Razavy equations. In [44] it is shown thatthe Razavy equation (4.12) is a quasi-exactly solvable algebraic equation that ad-mits n + 1 finite (polynomial times decaying exponential) solutions if M = n + 1and n ∈ N is a natural number. Moreover, there are n + 1 solutions for n + 1different values of λ characterized by an integer m = 1, 2, 3, · · · , n + 1. All the

eigenvalues between 0 and 2(1+δ)2

�2 of the supersymmetric modified Euler/Coulombspectral problem in SH0 are obtained in this way:

M = n+ 1 ⇒ E(0)n =

2(1 + δ)2

�2

(1− 1

(n+ 1)2

).

Concerning the eigenfunctions, we search for solutions of the M = n + 1 Razavyequations by means of the series expansion 9:

ηn(z) = z−n2 e−

ζn4 (z+ 1

z )∞∑k=0

(−1)k Pk(λ)

(2ζ)k k!zk , z = e2x , ζn =

4(1 + δ)

�2(n+ 1)

where Pk(λ) are polynomials of order k in λ to be fixed. The ODE Razavy equationis then solved if the following three-term recurrence relations among the polynomialshold:

Pk+1(λ) =(λ− (4k(n− k) + 2n+ 1 + ζ2n)

)Pk(λ)−(4k(n+1−k)ζ2n)Pk−1(λ) , k ≥ 0.

In particular, if λnm, m = 1, 2, · · · , n + 1, is one of the n + 1 roots of Pn+1,Pn+1(λnm) = 0, then

0 = Pn+2(λnm) = Pn+3(λnm) = Pn+4(λnm) = . . .

and the series truncates. The degeneracy in the energy is broken by the eigenvalues

I of the symmetry operator I(0)2 provided by the roots λnm in the form

Inm =�2

4(λnm − (n+ 1)2)− 4(1 + δ)2

�2,

which distinguishes between the different polynomials Pm(λnm).We solve the finite-step recurrences for the lower-energy cases

• n = 0:

ζ0 =4(1 + δ)

�2, P0(λ) = 1 , P1(λ) = λ− (1 + ζ20 ) , λ01 = 1 + ζ20

• n = 1:

ζ1 =2(1 + δ)

�2, P0(λ) = 1 , P1(λ) = λ− (3 + ζ21 )

P2(λ) = λ2 − 2(3 + ζ21 )λ+ 9 + 2ζ21 + ζ41

λ11 = 3− 2ζ1 + ζ21 , λ12 = 3 + 2ζ1 + ζ21

9This ansatz, and others related to this, allows to represent the Razavy Hamiltonians interms of differential operators that belong to the enveloping algebra of SL(2,R), see [44] and [45].


Table 4. Lower eigenvalues and eigenfunctions.

Energy Eigen-function

E(0)0 = 0 η

(0)01 (u) = e

− 2(1+δ)u

�2

E(0)1 =

3(1+δ)2

2�2

⎧⎪⎪⎪⎨⎪⎪⎪⎩η(0)11 (u) = e

− (1+δ)u

�2√

2(u + 1)

η(0)12 (u) = −e

− (1+δ)u

�2√

2(u − 1)

E(0)2 =

16(1+δ)2

9�2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

η(0)21 (u) = −2e

− 2(1+δ)u

3�2√

u2 − 1

η(0)22 (u) = 3�2

4(1+δ)e− 2(1+δ)u

3�2

[8(1+δ)

3�2 u − 1 +

√1 +

64(1+δ)2

9�4

]

η(0)23 (u) = 3�2

4(1+δ)e− 2(1+δ)u

3�2

[8(1+δ)

3�2 u − 1 −√

1 +64(1+δ)2

9�4

]

Separation constant

I(0)01 = 0

⎧⎪⎪⎨⎪⎪⎩I(0)11 = − �

2

4− 3(1+δ)2

�2 − (1 + δ)

I(0)12 = − �

2

4− 3(1+δ)2

�2 + (1 + δ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

I(0)21 = −�

2 − 32(1+δ)2

9�2

I(0)22 = − �

2

2− 32(1+δ)2

9�2 − �2

2

√1 +

64(1+δ)2

9�4

I(0)23 = − �

2

2− 32(1+δ)2

9�2 + �2

2

√1 +

64(1+δ)2

9�4

• n = 2

ζ2 =4(1 + δ)

3�2, P0(λ) = 1 , P1(λ) = λ− (5 + ζ22 )

P2(λ) = (λ− (5 + ζ22 ))(λ− (9 + ζ22 ))− 8ζ22

P3(λ) = (λ− (5 + ζ22 ))[(λ− (5 + ζ22 ))(λ− (9 + ζ22 ))− 16ζ22

]λ21 = ζ22 + 5 , λ22 = ζ22 + 7− 2

√1 + 4ζ22 , λ23 = ζ22 + 7 + 2

√1 + 4ζ22

and show the results for the lower eigenvalues and eigenfunctions in the Table 4.4.5.1. Contribution from the Whittaker-Hill equations. For these values of the

eigenvalues En and Inm two of the parameters of the Whittaker-Hill equations


(4.13) become:

βn = − 4

�2

√(1 + δ)2

(n+ 1)2− 4δ , μnm = N2

n +4

�2

(Inm +

4(1− δ)2

�2

)but the other one Nn = 1√

1−( 1+δ1−δ )

2(1− 1

(n+1)2

) is not a positive integer if n > 1 and

δ = Z2

Z1. The existence of solutions of the equation

Z1

[(k + 1)

√n(n+ 2)− (n+ 1)

√k(k + 2)

]= Z2

[(k + 1)

√n(n+ 2) + (n+ 1)

√k(k + 2)

]over positive integers n and k would be necessary to simultaneously find M =n + 1 and N = k + 1. It is clear that this is not the case and there are no finitesolutions other than the ground state in the supersymmetric two-center problem.The analogous equation in the non supersymmetric case is:

Z1 =n+ k + 2

n− k.Z2 ,

which admits solutions for positive integers n and k found by Demkov and hiscolleagues over forty years ago, see [43] and, e.g., [51].

If n = 0, E0 = 0, I01 = 0, however, the Whittaker-Hill equation is also QES,see [50], with a unique finite wave function:

β0 = − 4

�2(1− δ) , N0 = 1 , μ01 = 1 +

16(1− δ)2

�4, ξ

(0)01 (v) = e

2(1−δ)

�2 v .

The analytic wave function of the ground state in the SH0 sector is:(4.14)

E(0)0 = 0 , ψ

(0)01 (u, v) = η

(0)01 (u)ξ

(0)01 (v) = exp

[−2(1 + δ)u

�2

]· exp

[2(1− δ)v

�2

];

thus, there is a normalizable Bosonic zero energy ground state, a zero mode

ψ(0)01 (u, v), in the supersymmetric two-center system. Supersymmetry is not spon-

taneously broken.In the WH equation for the above parameters we could try a solution of the

form [50]:

ξn(w) = w1−Nn

2 e−βn4 (w+ 1

w )∞∑k=0

(−1)k Qk(μnm)

(2βn)k k!wk , w = e2iy ,

which solves (4.13) if the “recurrence” relations between the Q-polynomials hold:

Qk+1 =(μnm − (4k(Nn − 1− k) + 2Nn − 1 + β2

n))Qk − (4k(Nn − k)β2

n)Qk−1 k ≥ 0 .

This strategy, however, is not useful in this situation because the WH equationsare not QES if n > 1 (N �= k + 1).

Instead, we consider the WH equations in their algebraic form:

d2ξ(0)nm

dv2− v

1− v2dξ

(0)nm

dv+

Anm +Bnmv + Cnmv2

1− v2ξ(0)nm(v) = 0

Anm = −Inm�2− 4(1− δ)2

�4, Bnm =

2(1− δ)

�2, Cnm = − 2

�2

(En −

2(1− δ)2

�2

).


Now, following the standard theory of Hill equations, see [53], we search for powerseries solutions: if k ∈ N is a natural number,

ξ(0)nm(v) =

∞∑k=0

c(k)nm vk c

(2)nm = −Anm

2c(0)nm c

(3)nm = −1

6

((Anm − 1)c

(1)nm +Bnmc

(0)nm

)c(k)nm = − 1

k(k − 1)

{(Anm − (k − 2)2

)c(k−2)nm +Bnmc

(k−3)nm + Cnmc

(k−4)nm

}k ≥ 4 .(4.15)

We encounter fourth-term recurrence relations, a difficult situation to deal with,although the two basic solutions are easily characterized:

(1) c(0)nm+ = 1, c

(1)nm+ = 0: ξ

(0)nm+(v) = 1 +

∞∑k=2

c(k)nm+ vk.

(2) c(0)nm− = 0, c

(1)nm− = 1: ξ

(0)nm−(v) = v +

∞∑k=2

c(k)nm− vk.

These series converge in the open (−1, 1) interval and are extended to cover thesingularities v = ±1 setting the values:

ξ(0)nm+(1) = 1 +

∞∑k=2

c(k)nm+ , ξ

(0)nm+(−1) = 1 +

∞∑k=2

(−1)k c(k)nm+(4.16)

ξ(0)nm−(1) = 1 +

∞∑k=2

c(k)nm− , ξ

(0)nm−(−1) = −1 +

∞∑k=2

(−1)k c(k)nm− .

The series for the ground state, for instance, are easy to find:

ξ(0)01 (v) = exp

[2(1− δ)

�2v

]=

∞∑k=0

c(k)01 vk c

(k)01 =

1

k!

(2(1− δ)

�2

)k

c(0)01 = ξ

(0)01 (0) = 1

dξ(0)01

dv(v) =

2(1− δ)

�2exp

[2(1− δ)

�2v

]=

∞∑k=1

kc(k)01 vk−1 , c

(1)01 =

dξ(0)01

dv(0) =

2(1− δ)

�2

ξ(0)01 (±1) =

∞∑k=0

(±1)k

k!

(2(1− δ)

�2

)k

= exp

[±2(1− δ)

�2

].

Any other solution of the WH equations is obtained by specific linear combinationsof the two basic solutions. We will choose linear combinations of the general form

(4.17) ξ(0)nm(v) = c+ ξ(0)nm+(v) + c− ξ

(0)nm−(v)

In fact, any choice of (c+, c−) ∈ C2 in (4.17) fixes the extension to the bound-

ary of the elliptic strip E = [1,+∞) × [−1, 1] of the wave functions ψ(0)nm(u, v) =

η(0)nm(u)ξ

(0)nm(v). Id est, extensions of H0 in L2(E) are determined from the values

of ψ(0)nm(u, v) at the boundary: ψ

(0)nm(1, v) and ψ

(0)nm(u,±1). We remark that these

extensions are not essentially self-adjoint in general; different eigenfunctions havea small overlap for generic values of �.

4.6. Two centers of the same strength. In the case δ = 1 when thestrength of the centers is identical the supersymmetric spectral problem in SH0simplifies remarkably:[

−�2(u2 − 1)d2

du2− �2u

d

du+

(16

�2(u2 − 1)− 4u− 2Eu2

)]η(0)E (u) = Iη

(0)E (u)(4.18) [

−�2(1− v2)d2

dv2+ �2v

d

dv+ 2Ev2

]ξ(0)E (v) = −Iξ

(0)E (v) .(4.19)


Again, (4.18) is equivalent to a Razavy equation

(4.20) −d2η(0)E (x)

dx2+ (ζ cosh 2x−M)2 η

(0)E (x) = λ η

(0)E (x) ,

with parameters

ζ =2

�

√16

�2− 2E , M2 =

8

8− �2E, λ = M2 +

4

�2

(I +

16

�2

)after the change of coordinates: u = cosh2x.

The v-equations (4.19), however, become the Mathieu equations

(4.21) −d2ξ(0)E (y)

dy2+ (α cos4y + σ) ξ

(0)E (y) = 0

with parameters

α =4E

�2, σ =

4

�2(I + E) ,

under the change: v = cos2y. The strategy to solve these two entangled equationsis the same as in the case of two centers of different strength.

First, we search for finite solutions of the Razavy equation. The procedure isidentical and we only need to replace δ by 1 in the formulae of the previous Section§. 4.5. We now have

E(0)n =

8

�2

(1− 1

(n+ 1)2

), ζn =

8

�2(n+ 1), Inm =

�2

4(λnm−(n+1)2)−16

�2

where λnm,m = 1, 2, · · · , n+ 1 are the roots of the polynomials Pn+1(λ) that cutthe series. We thus show the results for the lower eigenvalues and eigenfunctions inthe next Table:

Energy Eigen-function Separation constant

E(0)0 = 0 η

(0)01 (u) = e

− 4u�2 I

(0)01 = 0

E(0)1 = 6

�2

⎧⎪⎪⎨⎪⎪⎩η(0)11 (u) = e

− 2u�2

√2(u + 1)

η(0)12 (u) = −e

− 2u�2

√2(u − 1)

⎧⎪⎪⎨⎪⎪⎩I(0)11 = − �

2

4− 12

�2 − 2

I(0)12 = − �

2

4− 12

�2 + 2

E(0)2 = 64

9�2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

η(0)21 (u) = −2e

− 4u3�2

√u2 − 1

η(0)22 (u) = 3�2

8e− 4u

3�2[

163�2 u − 1 +

√1 + 256

9�4

]η(0)23 (u) = 3�2

8e− 4u

3�2[

163�2 u − 1 −

√1 + 256

9�4

]

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

I(0)21 = −�

2 − 1289�2

I(0)22 = − �

2

2− 128

9�2 − �2

2

√1 + 256

9�4

I(0)23 = − �

2

2− 128

9�2 + �2

2

√1 + 256

9�4

Nothing new with respect to the non equal centers case.4.6.1. Contribution from the Mathieu equations. The novelty comes from the

Mathieu equations: unlike the Whittaker-Hill equations these equations are neverquasi-exactly solvable -there are no finite solutions whatsoever- but, instead thereare lots of solutions that can be described analytically in terms of the Mathieu sineand cosine special functions.


As in the non-equal centers system, the ground state is an exception. It is notruled by any Mathieu equation. α0 = 4

�2E0 = 0, σ01 = 4�2 (I01 + E0) = 0 implies

that:

−d2ξ(0)01

dy2= 0 ⇒ ξ

(0)01 (y) = Ay +B .

If the two centers have the same strength an important discrete symmetry arises:the r1 ↔ r2 exchange is not detectable. v ↔ −v, or, y ↔ y + π

2 is a symmetry of

the system and we remain with the only invariant solutions ξ(1)01 (y) = B under this

transformation. There are only two independent choices: B = 1 and B = 0. Thefirst choice is “even” in v, the second choice is “odd” in v but negligible. The zeroenergy ground state is therefore built from the even v-independent wave function:

ψ(0)01 (u, v) = η

(0)01 (u)ξ

(0)01 (v) = e−

4u�2 .

The parameters of the Mathieu equations determined by the spectral problemfor positive energy (n ≥ 1) are:

αn = 4En

�2, σnm = 4

En + Inm�2

.

The v ↔ −v symmetry of equation (4.19) is translated into the y ↔ y + π2n,

n ∈ Z, infinite discrete symmetry of the Mathieu equation. The Mathieu cosineand Mathieu sine special functions are obtained from the Bloch-type solutions ofthe Mathieu equation ruled by the discrete translational symmetry with Floquetindices determined from the parameters α and σ. Because the Mathieu equation is

blind to the v ↔ −v exchange, if ξ(0)nm(v) is a solution ξ

(0)nm(−v) also solves (4.19). In

[39] we chose even and odd in v combinations of the Mathieu functions, a situationclosely related to the hidden quantum supersymmetry in Bosonic systems unveiledin [56]. Instead, here we choose the combination

(4.22) ξ(0)nm(v) = C [anm, qn, arccos(v)] + iS [anm, qn, arccos(v)] ,

to build positive energy eigenfunctions, a choice designed to go to the Kepler/Cou-lomb system at the d = 0 limit. Here, C[a, q, z] and S[a, q, z] are respectively thecosine and sine Mathieu functions and qn = αn

8 , anm = −σnm

4 .We now list some of the lower values of the parameters

q1 = 3�4 ,

{a11 = 6

�4 + 2�2 + 1

4

a12 = 6�4 − 2

�2 + 14

q2 = 329�4 ,

⎧⎨⎩a21 = 1 + 649�4 ,

a22 = 12 + 64

9�4 + 12

√1 + 256

9�4

a23 = 12 + 64

9�4 − 12

√1 + 256

9�4

and offer a Table 5 showing the probability densities of some eigenfunctions forseveral values of �. For instance, � = 0.7 is the value of this non-dimensionalparameter for the hydrogen molecule ion. � = 114, 7 corresponds to the ionizedhelium atom; note that the radius of the nuclei is of the order of 10−15cm, etcetera.One notices that the smaller � is, the more classical is the system, the wave functionsbeing more concentrated around the centers.


Table 5. N = 2 supersymmetric wave functions in SH0 of twoequal centers: δ = 1

|ψ(0)nm(x1, x2)|2 � = 0.7 � = 1 � = 2 � = 4

n=0, m=1

n=1, m=1

n=1, m=2

n=2, m=1

n=2, m=2

n=2, m=3


4.7. Comparison between the N = 2 supersymmetric and the N = 0spectra. The energy spectra of the Euler Hamiltonian - given by the solution ofthe system of equations (4.2), (4.3)- and the H0 Euler supersymmetric Hamiltonianreduced to the SH0 sector of the supersymmetric Hilbert space -characterized bythe system (4.10), (4.11)- are listed below:

N = 0 , N = 2

0 < δ = Z2

Z1≤ 1 En = − 2(1+δ)2

�2(n+1)2, E

(0)n = 2(1+δ)2

�2

(1− 1

(n+1)2

)The degeneracy of the nth level is n+1 in both cases and it is split by the eigenvaluesof the symmetry operator, respectively I and I(0).

As in the supersymmetric Kepler/Coulomb problem there are bound states inSH1 of the form:

ψ(1)nm(r1, r2) = Q† ψ(0)

nm(r1, r2) , n ≥ 1 , m = 1, 2, · · · , n+ 1 .

Unlike in the supersymmetric Kepler problem, there is one zero energy wave func-tion in SH1. Coming back to dimension-full coordinates and parameters, it is:

Qψ(1)0 (x1, x2) = Q†ψ

(1)0 (x1, x2) = 0 , E

(1)0 = 0

ψ(1)0 (x1, x2) =

⎛⎜⎜⎝0

∓ r1+r24dr1r2

√4d2 − (r1 − r2)2

r2−r14dr1r2

√(r1 + r2)2 − 4d2

0

⎞⎟⎟⎠ e−2mα(r2+δr1)

�2 ;(4.23)

here, the “− ” sign occurs for x2 > 0 whereas the “+” sign arises when x2 < 0. Inthe limit where the two centers coincide this wave function becomes:

limd→0

ψ(1)0 (x1, x2) =

⎛⎜⎜⎝0

− x2

x21+x2

2x1

x21+x2

2

0

⎞⎟⎟⎠ e−2mα(1+δ)

�2

√x21+x2

2

and we observe that its norm diverges:

limd→0

∫ ∞

−∞

∫ ∞

−∞dx1 dx2

∣∣∣ψ(1)0 (x1, x2)

∣∣∣2 =

= limd→0

2πK0

(4mαd

�2(1 + δ)

)I0

(4mαd

�2(1− δ)

)= +∞ ,

where K0(z) and I0(z) are respectively the first-order modified Bessel functions,see [18]. Thus, we confirm that there is no fermionic zero mode in the SUSYKepler problem. Note that ground states in that problem must live in the scalarrepresentation of SO(3).

The comparison between the Bosonic and Fermionic zero modes can be seen inTable 6, where the center on the right is twice as strong as the center on the left andδ = 1

2 . In the Bosonic ground state the electron is concentrated around the stronger

center for small � but becomes spread over the two centers when � increases. In theFermionic ground state the superparticle behaves in the opposite way!: for small� it is concentrated in the weaker center and two peaks on the two centers arisefor larger �. Moreover, for any value of � the probability density of the groundstate in SH1 is always peaked at the centers. In any other respect, i.e., concerning


Table 6. N = 2 supersymmetric zero modes.

Probabilityδ= 1

2

|ψ(1)01 (x1,x2)|2

δ= 12

|ψ(1)01 (x1,x2)|2

δ=1

|ψ(1)01 (x1,x2)|2

density

� = 0.7

� = 1

� = 2

� = 4

the scattering solutions with energy greater than 2�2 (1 + δ)2, the structure of the

spectrum is qualitatively identical to the spectrum of the supersymmetric Keplerproblem. There are scattering states in SH0 paired via the Q† supercharge toscattering states in SH1 and scattering states in SH2 paired via the Q superchargeto scattering states in SH1. All this is depicted schematically in Figure 3:

5. Two center collapse in one center

In this Section we shall analyze how the two main spectral problems describedrespectively in sections §.3 and §.4 are connected. The link appears at the d = 0limit of the two-center problem. In order to go to this singular limit of the modifiedEuler/Coulomb problem it is necessary to restore full dimensional variables. Thus,with no change of notation the parameters and physical variables to be dealt within this Section have the proper dimensions. We shall perform the limit in the caseδ = 1 because we have full analytical information for two equal centers.


Figure 3. The spectrum of the Euler Hamiltonian (left panel).The spectrum of the supersymmetric Euler Hamiltonian (rightpanel).

• We start from the ground state characterized by: n = 0 and m = 1. Theeigenvalues of the Hamiltonian and the symmetry operator, the Mathieu parame-ters, and the ground state wave function, are respectively:

E(0)0 = 0 I01 = 0 q0 = 0 a01 = 0 ψ

(0)01 (u, v) =

(mα

�2

)e−

4mα�2 u

At the d→ 0 limit we have:

r = limd→0

u =√x21 + x2

2 ϕ = limd→0

arccos(v

d) = arctan(

x2

x1) .

Therefore,

ψ(0)00 (r, ϕ) = lim

d→0ψ(0)01 (u, v) =

(mα

�2

)e−

2mα�2 r , α = 2α

and the ground state of the one-center problem with twice the electric charge ap-pears at the d = 0 limit.• We next consider the doublet labeled by n = 1, m = 1 and m = 2. The

eigenvalues of the Hamiltonian and symmetry operators are:

E(0)1 =

6mα2

�2,

I11 = −�2

4 − 2(mdα)− 12(mdα)2

�2

I12 = −�2

4 + 2(mdα)− 12(mdα)2

�2

The corresponding Mathieu parameters and eigenfunctions read:

q1 =3(mdα)2

�4,

a11 = 14 + 2 (mdα)

�2 + 6(mdα)2

�4

a12 = 14 − 2 (mdα)

�2 + 6(mdα)2

�4

ψ(0)11 (u, v) =

(mα

�2

)3/2e− 2mα

�2 u√2(u+ d)

{C[a11, q1, arccos(

v

d) + iS[a11, q1, arccos(

v

d)]}

ψ(0)12 (u, v) = −

(mα

�2

)3/2e− 2mα

�2 u√2(u− d)

{C[a12, q1, arccos(

v


v

d)]}


Because

limd→0

q1 = 0 α1k = limd→0

a1k =1

4k = 1, 2 , lim

d→0I11 = lim

d→0I12 = −�2

4

cos√α1k ϕ = lim

d→0C[a1k, q1, arccos(

v

d)] sin

√α1k ϕ = lim

d→0S[a1k, q1, arccos(

v

d)]

we find

ψ(0)12

12

(r, ϕ) = limd→0

ψ(0)11 (u, v) =

(mα

�2

)3/2

r1/2 e−mα�2 r ei

12ϕ

ψ(0)12−

12

(r, ϕ) = limd→0

ψ(0)12 (u, v) = −

(mα

�2

)3/2

r1/2 e−mα�2 r e−i 1

2ϕ .

Again we find that the eigenvalue of the Hamiltonian operator remains the sameat the d = 0 limit, the eigenvalues of the symmetry operator go to the eigenvaluesof the square of the angular momentum, and the wave functions become the eigen-functions of the Kepler/Coulomb problem with spin of one-half. Note that we have

used:√

14 = ± 1

2 .

• Finally, we consider the triplet n = 2, m = 1, m = 2 and m = 3. Theeigenvalues of the Hamiltonian and the symmetry operator are:

E(0)2 =

64mα2

9�2,

I21 = −�2 − 128(mdα)2

9�2

I22 = −�2

2 −128(mdα)2

9�2 − �2

2

√1 + 256(mdα)2

9�2

I23 = −�2

2 −128(mdα)2

9�2 + �2

2

√1 + 256(mdα)2

9�2

whereas the Mathieu parameters read:

q2 =32(mdα)2

9�4,

a21 = 1 + 64(mdα)2

9�4

a22 = 12 + 64(mdα)2

9�4 + 12

√1 + 256(mdα)2

9�4

a23 = 12 + 64(mdα)2

9�4 − 12

√1 + 256(mdα)2

9�4

.

The eigenfunctions are more complicated

ψ(0)21 (u, v) =

(mα

�2

)2e− 4mα

3�2 u√

u2 − d2{C[a21, q2, arccos(

v


v

d)]}

ψ(0)22 (u, v) =

(mα

�2

)2 3�2

8mαe− 4mα

3�2 u

⎡⎣16mα

3�2u− 1 +

√1 +

256(mdα)2

9�2

⎤⎦×{C[a22, q2, arccos(

v


v

d)]}

ψ(0)23 (u, v) =

(mα

�2

)2 3�2

8mαe− 4mα

3�2 u

⎡⎣16mα

3�2u− 1−

√1 +

256(mdα)2

9�2

⎤⎦×{C[a23, q2, arccos(

v


v

d)]}


but the limits are similar:

limd→0

q2 = 0 α21 = limd→0

a21 = 1 = limd→0

a22 = α22 α23 = limd→0

a23 = 0

limd→0

I21 = limd→0

I22 = −�2 limd→0

I23 = 0 k = 1, 2, 3

cos√α2k ϕ = lim

d→0C[a2k, q2, arccos(

v

d)] sin

√α2k ϕ = lim

d→0S[a2k, q2, arccos(

v

d)] .

Thus, we find

ψ(0)11 (r, ϕ) = lim

d→0ψ(0)21 (u, v) =

(mα

�2

)2

r e−2mα3�2 r eiϕ

ψ(0)1,−1(r, ϕ) = lim

d→0ψ(0)22 (u, v) =

(mα

�2

)2

r e−2mα3�2 r e−iϕ

ψ(0)10 (r, ϕ) = lim

d→3ψ(0)23 (u, v) =

(mα

�2

)2 (3�2

mα− 4r

)e−

2mα3�2 r

again falling in wave functions of one doubly charged center, in this case withangular momentum 1.

3D graphics of this analysis are shown in Table 7. We remark that the graphicsare drawn in non-dimensional variables and some of them cannot skip the 1 in thed = 0 limit displaying non-smooth tendency to the Kepler wave functions; namelythe n = 2,m = 1 and n = 2,m = 2 wave functions.

5.1. Collapse of two centers of different strength. When the two centershave different strength, δ �= 1, the one center collapse happens exactly in thesame way as compared to the equal two-center collapse because limd=0 Bnm =limδ=1 Bnm = 0. Id est, when the charges are superposed the only things thatmatter is the total charge. We now offer this analysis for completeness which willshed light on the physical nature of the recurrence relations (4.15) and the basicsolutions (4.16) of the Whittaker-Hill equations. Because we prefer to deal withpower series with purely numerical coefficients we shall stick to non-dimensionalv-variables and put all the physical dimensions back in the u-dependent part of thewave functions. We recall that:

Anm = − 1

�2

(Inm +

4(1− δ)2

�2

), Bnm =

2(1− δ)

�2, Cnm = − 2

�2

(En −

2(1− δ)

�2

)are the non-dimensional parameters of the algebraic WH equations.

• Start with the ground state: n = 0, m = 1. The values A01 = −(

2(1−δ)

�2

)2

,

B01 = 2(1−δ)�2 , and C01 = 4(1−δ)2

�4 means that the recurrence relations (4.15) are

solved by c(k)01 = 1

k!

(2(1−δ)

�2

)k

. Thus, ξ(0)01 (v) =

∑∞k=0

1k!

(2(1−δ)

�2

)k

vk must be

multiplied by the unique finite solution of the E0 = I01 = 0 Razavy equation:

ψ(0)01 (u, v) =

(mα(1 + δ)

�2

)e−

2mα(1+δ)

�2 u e2mα(1−δ)

�2 v .

It is clear, like above, that now

limd→0

ψ(0)01 (u, v) =

(mα

�2

)e−

2mα�2 r = ψ

(0)00 (r, ϕ) , α = (1 + δ)α


Table 7. 3D Plots of the probability densities: Kepler versus Euler.

Probability Kepler Two Centers

density � = 1 � >> 1 � >> 1

j=0 m=0n=0 m=1

j= 12 m=± 1

2n=1 m=1,2

j=1 m=±1n=2 m=1,2

j=1 m=0n=2 m=3

and we find the ground state of only one-center with zero energy and momentumand electric charge (1 + δ)α.

• Next, we consider the doublet: n = 1 ,

{m = 1m = 2

. The eigenfunctions are

of the form

ψ(0)11 (u, v) =

(mα(1 + δ)

�2

)3/2

e−m(1+δ)α

�2 u√2(u+ d) ξ

(0)11 (v)

ψ(0)12 (u, v) = −

(mα(1 + δ)

�2

)3/2

e−m(1+δ)α

�2 u√2(u− d) ξ

(0)12 (v)

where the ξ(0)11 (v) and ξ

(0)12 (v) are linear combinations of the basic solutions of the

WH equations to be identified in such a way that the Kepler/Coulomb wave func-

tions of energy E1 = 3mα2

�2 and angular momentum �

2 and −�

2 arise at d = 0.The pass to the limit is identical to the collapse of two equal centers in the

u-part of the wave function. Therefore, we will describe in detail only the collapse


of the v-part. Because the d = 0 limits of the WH parameters are

limd→0

A11 = limd→0

A12 =1

4, limd→0

B11 = limd→0

B12 = 0 = limd→0

C11 = limd→0

C12

the recurrence relations become very easy: if we denote q = 1, 2,

c(2)1q = −1

8c(0)1q c

(3)1q = −1

6

(1

4− 1)

)c(0)1q c

(k)1q = − 1

k(k − 1)

(1

4− (k − 2)2

)c(k−2)1q .

The basic solutions, corresponding respectively to the choices c(0)1q+ = 1, c

(1)1q+ = 0

and c(0)1q− = 0, c

(1)1q− = 1, are:

c(2)1q+ = −1

8, c

(4)1q+ = − 5

128, c

(6)1q+ =

21

1024, · · · , ξ

(0)1q+(v) =

∞∑k=0

c(2k)1q+v2k

c(3)1q− =

1

8, c

(5)1q− =

7

128, c

(7)1q− =

33

1024, · · · , ξ

(0)1q−(v) =

∞∑k=0

c(2k+1)1q− v2k+1 .

We know that C[ 14 , 0, arccosv]± iS[ 14 , 0, arccosv] goes to exp[±iϕ2 ] at d = 0. Com-parison with the power series expansion

C[1

4, 0, arccosv] + iS[

1

4, 0, arccosv] =

=1 + i√

2+

(12 −

i2

)v√

2−

(18 + i

8

)v2√

2+

(116 −

i16

)v3√

2−

(5

128 + 5i128

)v4√

2

+

(7

256 −7i256

)v5√

2−

(21

1024 + 21i1024

)v6√

2+O

(v7)

and the analogous series with the relative minus sign, tells us that the right com-

binations such that limd→0 ξ(0)11 (v) = exp[iϕ2 ] and limd→0 ξ

(0)12 (v) = exp[−iϕ2 ] are:

ξ(0)1q (v) =

1− i(−1)q√2

ξ(0)1q+ +

1 + i(−1)q

2√2

ξ(0)1q− .

Henceforth,

limd→0

ψ(0)11 (u, v) = lim

d→0η(0)11 (u)ξ

(0)11 (v) =

(mα

�2

)3/2

r1/2 e−mα�2 r ei

12ϕ = ψ

(0)12

12

(r, ϕ)

limd→0

ψ(0)12 (u, v) = lim

d→0η(0)12 (u, v)ξ

(0)12 (v) = −

(mα

�2

)3/2

r1/2 e−mα�2 r e−i 1

2ϕ = ψ(0)12−

12

(r, ϕ)

• Finally, we address the triplet state n = 1 ,

⎧⎨⎩m = 1m = 2m = 3

. The d = 0 limits of

the WH parameters are in this case

limd→0

A21 = limd→0

A22 = 1 , limd→0

A23 = 0 , limd→0

B2q = 0 = limd→0

C21q , q = 1, 2, 3 .


The basic solutions for q = 1, 2 are:

c(0)2q+ = 1 , c

(1)2q+ = 0 , c

(2)2q+ = −1

2, c

(4)2q+ = −1

8, c

(6)2q+ = − 1

16, c

(8)2q+ = − 5

128· · ·

c(0)2q− = 0 , c

(1)2q− = 1 , c

(k)2q− = 0, ∀k ≥ 2 , · · ·

ξ(0)2q+(v) =

∞∑k=0

c(2k)1q+v2k , ξ

(0)2q−(v) =

∞∑k=0

c(2k+1)1q− v2k+1 = v .

Comparison with the power series expansion

C[1, 0, arccosv]+ iS[1, 0, arccosv] = i+v− iv2

2− iv4

8− iv6

16− 5iv8

128− 7iv10

256+O

(v11

)show us that the combinations

ξ(0)21 (v) = iξ

(0)21+(v) + ξ

(0)21−(v) ξ

(0)22 (v) = −iξ(0)22+(v) + ξ

(0)22−(v)

go respectively to exp[iϕ] and exp[−iϕ] at the d = 0 limit.The recurrence relations for n = 2, m = 3 are solved by

c(0)23+ = 1 , c

(1)23+ = 0 , c

(k)23+ = 0 , ∀k ≥ 2 , ξ

(0)23+(v) = 1

c(0)23− = 0 , c

(1)23− = 1 , c

(3)23− =

1

6, c

(5)23− =

3

40, c

(7)23− =

5

112,

· · · ξ(0)23−(v) =∞∑k=0

c(2k+1)1q− v2k+1.

Because C[0, 0, arccosv] + iS[0, 0, arccosv] = 1√2the linear combination ξ

(0)23 (v) =

1√2ξ(0)23+(v) leads to the Kepler/Coulomb eigenfunction in the d = 0 limit.

In fact, we obtain in this limit the right Kepler/Coulomb triplet with en-

ergy E(0)2 = 16mα

9�2 and angular momenta 1 - limd→0 I21 = limd→0 I22 = −�2,limd→0 I23 = 0 - :

limd→0

ψ(0)21 (u, v) = lim

d→0η(0)21 (u)ξ

(0)21 (v) =

(mα

�2

)2

r e−2mα3�2 r eiϕ = ψ

(0)11 (r, ϕ)

limd→0

ψ(0)22 (u, v) = lim

d→0η(0)22 (u)ξ

(0)22 (v) =

(mα

�2

)2

r e−2mα3�2 r e−iϕ = ψ

(0)1−1(r, ϕ)

limd→0

ψ(0)23 (u, v) = lim

d→0η(0)23 (u)ξ

(0)23 (v) =

(mα

�2

)2 (3�2

mα− 4r

)e−

2mα3�2 r = ψ

(0)10 (r, ϕ)

6. Further comments

We end this long survey by thinking about further possible extensions of theseideas and calculations to other classical integrable systems.

(1) The immediate temptation is to address the Kepler/Coulomb problem inRN . The N -dimensional supersymmetric Kepler/Coulomb problem hasalready been developed and fully solved in [32]. In our approach it would

be easily doable in the N = 0/N = 1 and N = N/N = N − 1 sectors;the wave functions in L2(R+ × SN−1) would be organized in irreduciblerepresentations of SO(N+1) -the bound states- or SO(N, 1) -the scatteringstates-. In both cases, the wave functions are Kummer confluent functionstimes the spherical harmonics in L2(SN−1). Of course, the structure of the


supersymmetric Hilbert space is more complicated; there are N + 1 sub-

spaces of dimension

(Nk

), k = 0, 1, · · ·N , so that:

∑Nk=0

(Nk

)= 2N .

This means that in the k-subspaces such that 2 ≤ k ≤ N − 2 only theprocedure proposed in [32] would be effective.

The supersymmetric modified Euler/Coulomb problem in RN doesnot pose more difficulties than those encountered in this paper becauseall the N − 2 additional variables are cyclic.

(2) It will also be interesting to build the supersymmetric extension of theKepler/Coulomb problem constrained to a sphere. This problem in thenon supersymmetric framework was addressed independently by Higgsand Pronko in [57] and [58]. The supersymmetric generalization willrequire to deal with all the subtleties of spinors living in non-flat manifolds.The metric, the N -bein, the spin connection, and the like will enter thesupercharges one way or another to make the system more intricate, seee.g. [59] where supersymmetric systems in curvilinear coordinates areconstructed.

The Euler problem considered on a sphere, see [60], is also a fairly wellknown integrable system with applications in celestial mechanics. It seemspromising and interesting to work out the corresponding supersymmetricextension.

(3) Another important integrable system is the Neumann problem [61]: aparticle forced to move on a sphere under the action of attractive elasticforces. This has been a source of inspiration for treatises on dynami-cal integrable systems, see [62] and [63], and it has been applied, in therepulsive case, by some of us to study topological defects in non-linearsigma models with quadratic [64] and quartic [65] potentials. We believethat the supersymmetric extension of the quantum version of the Neu-mann problem will provide a physical example of the systems envisagedby Witten in [5] to construct a quantum derivation of Morse theory.

(4) The Bosonic zero modes ψ(0)00 (r, φ) in (3.2) and ψ

(0)01 (u, v) in (4.14) have

been easy to find. The Fermionic zero mode ψ(1)0 (x1, x2) (4.23) in the two

center problem was discovered by means of supercharges defined in ellipticcoordinates, translated back to Cartesian coordinates, and checked withina Mathematica environtment. A Fermionic zero mode in the one centerproblem, however, does not exist [2], [32]. This is very intriguing andcompels us to study SUSY quantum mechanics in curvilinear coordinatesin a more profound way. The way forward to and backward from thecurvilinear to Cartesian coordinates of all these structures is highly non-trivial, see [59] for early attempts in this program. We think that thedifficulties with the zero modes have a similar origin to the subtletiesarising in the definition of spinors on curved manifolds. We plan to analyzethis issue in a future publication.


Acknowledgements

All of us warmly acknowledge Mikhail Ioffe for lectures, seminars, talks, andconversations on Supersymmetric Quantum Mechanics over recent years. His mas-tery of supersymmetric QM has greatly helped us in the struggle to improve ourunderstanding of a matter with so many facets.

JMG is also indebted to Andreas Wipf for patiently hearing from him about thetwo centers problem and for the crucial suggestion of studying the Euler problemin close comparison with the Kepler problem. Besides being a very fruitful idea, itshowed us how to shape the structure of the paper.

We also thank Primitivo Acosta-Humanez, David Blazquez, Mayerling Nunez-Portela and Mikhail Plyuschay for giving us the opportunity to present this materialat the Jairo Charris Seminar 2010 Algebraic aspects of Darboux transformations,quantum integrable systems, and supersymmetric quantum mechanics, Santa Marta,Colombia, 4-7 August 2010. Part of this material was presented almost immediatelybefore in the Workshop Supersymmetric quantum mechanics and quantum spectraldesign, 18-28 July 2010, Benasque, Spain. We warmly thank the Benasque orga-nizers Alexander Andrianov, Luismi Nieto, and Javier Negro as well for their kindinvitation to participate. The two workshops, from the Aneto peak in the Pyreneesto the Sierra Nevada de Santa Marta on the Caribbean coast, were indeed verystimulating meetings on closely related subjects celebrated in rapid succession!!

Finally, we gratefully acknowledge that this work has been partially financed bythe Spanish Ministerio de Educacion y Ciencia (DGICYT) under grant: FIS2009-10546.

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Departamento de Matematica Aplicada and IUFFyM

Current address: Universidad de Salamanca, SPAINE-mail address: [email protected]

Departamento de Fisica and IUFFyM


Departamento de Matematicas


Departamento de Fisica and IUFFyM



Solvable rational extension of translationally shape invariantpotentials

Yves Grandati and Alain Berard

Abstract. Combining recent results on rational solutions of the Riccati-Schrodinger equations for shape invariant potentials to the scheme developedby Tkachuk, Fellows and Smith in the case of the one-dimensional harmonic

oscillator, we show that it is possible to generate an infinite set of solvable ra-tional extensions for every translationally shape invariant potential of secondcategory.

Contents

Introduction1. Solvable rational extensions of the harmonic oscillator2. Second category potentials3. ConclusionAcknowledgmentsReferences

Introduction

In quantum mechanics there exists only few families of potentials which areexactly solvable in closed-form. Most of them belong to the class of shape-invariantpotentials [1, 2, 3]. A possible way to generate new solvable potentials is to startfrom the known ones and to construct regular rational extensions of them. If theprocedure has a long history, in the last years important progress have been madein this direction [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. A nice example of sucha rational extension is provided by the so-called CPRS potential [14] which is arational extension of the one-dimensional harmonic oscillator. Very recently Fellowsand Smith [15] showed that this potential can be obtained as a supersymmetricpartner of the harmonic oscillator. In the same way they show how to generatean infinite family of partner potentials which are regular rational extensions of theharmonic oscillator. This partnership is based on the use of excited states Riccati-Schrodinger (RS) functions as superpotentials. This technique was devised for the

2010 Mathematics Subject Classification. Primary: 33C47, 81Q60; Secondary: 34A25,34B05, 81Q05.

Key words and phrases. Supersymmetric quantum mechanics, solvable quantum systems,exceptional orthogonal polynomials, Darboux transformation.


115

116 YVES GRANDATI AND ALAIN BERARD

first time by Robnik [16, 17], but the potentials obtained are singular. Fellows andSmith circumvent the problem by using a ”spatial Wick rotation” which eliminatesthe singularities from the real axis. This device was already suggested by Shnol’[18] in 1994 as a way to generate rational extensions of the harmonic potential byapplication of successive Darboux transforms. Four years later, Tkachuk [19] usedit to build families of exactly solvable potentials and in fact his work contains theessential of the results rediscovered by Fellows and Smith.

In a recent work [21] we propose a general scheme to obtain rational solutions tothe Riccati-Schrodinger equations associated to the whole class of translationallyshape invariant potentials. These last are shared into two categories which arerelated via simple changes of variables respectively to the harmonic oscillator andto the isotonic oscillator. In this letter we show how, by combining these results withthe Robnik-Tkachuk-Fellows-Smith (RTFS) technique, we can generate an infiniteset of regular rationally-extended solvable potentials from every shape invariantpotential of the second category in a very direct and systematic way.

1. Solvable rational extensions of the harmonic oscillator

1.1. Basic scheme. We recall the basic features of the RTFS technique [16,19, 15], putting it in a general form appropriate for what follows.

Let H = −d2/dx2 + V (x) of associated spectrum (En, wn) ≡ (En, ψn), wherewn(x) = −ψ′

n(x)/ψn(x). The Riccati-Schrodinger (RS) equation [21] for the levelEn is:

(1) −w′n(x) + w2

n(x) = V (x)− En,

where we suppose E0 = 0.Make a ”spatial Wick rotation”, that is, set x → ix, and define vn(x) =

−iwn(ix) Eq(1) becomes:

(2) v′n(x) + v2n(x) = V (n)(x),

where the ”Wick rotated” potential is given by:

(3) V (n)(x) = En − V (ix).

V (n)(x) is supposed to be real and to have no movable (that is n dependent) sin-gularity on the real line. Considering vn(x) as superpotential, V

(n) can be viewed

as the SUSY partner [1] of V (n) defined as:

(4) V (n)(x) = V (n)(x)− 2v′n(x) = V (ix)− En + 2v2n(x).

The positive hamiltonians H(n) and H(n), associated to V (n)(x) and V (n)(x)respectively, can be written:

(5)

{H(n) = A(n)+A(n)

H(n) = A(n)A(n)+,

with A(n) = d/dx+ vn(x).

If ψ(n)0 (x) ∼ exp

(−∫vn(x)dx

)is normalizable, it satisfies A(n)ψ

(n)0 = 0 and is

then the zero-energy ground state of H(n). In this case the two hamiltonians arealmost isospectral, that is:

(6)

{E

(n)0 = 0

E(n)k = E

(n)k+1, k ≥ 0,

SOLVABLE RATIONAL EXTENSION 117

and their eigenstates are related by:

(7)

⎧⎪⎨⎪⎩ψ(n)k (x) = 1√

E(n)k+1

A(n)ψ(n)k+1 (x)

ψ(n)k (x) = 1√

E(n)k

A(n)+ψ(n)k (x) .

In the case where ψ(n)0 (x) and 1/ψ

(n)0 (x) do not belong to the considered space

of bound states, the two hamiltonians are strictly isospectral, that is:

(8) E(n)k = E

(n)k , k ≥ 0,

and their eigenstates are related by:

(9)

⎧⎪⎨⎪⎩ψ(n)k (x) = 1√

E(n)k

A(n)ψ(n)k (x)

ψ(n)k (x) = 1√

E(n)k

A(n)+ψ(n)k (x)

.

Suppose that the potential considered satisfies the following identity (this isthe case of the harmonic and isotonic potentials):

(10) V (ix) = − (V (x) + δ) .

We then have:

(11) V (n)(x) = V (x) + δ + En, V (n)(x) = 2v2n(x)− V (x)− (En + δ) .

The spectrum of H(n) is:

(12) E(n)k = Ek + En + δ; ψ

(n)k (x) = ψk (x) ∼ exp

(−∫

wk(x)dx

), k ≥ 0.

As for the spectrum of H(n), it is either:

(13) E(n)k = Ek + En + δ; ψ

(n)k (x) = (Ek + En + δ)−1/2 A(n)+ψk (x) , k ≥ 0,

in the strictly isospectral case or:(14){

E(n)0 = 0; ψ

(n)0 (x) ∼ exp

(−∫vn(x)dx

)E

(n)k+1 = Ek + En + δ; ψ

(n)k+1 (x) = (Ek + En + δ)

−1/2A(n)+ψk (x) , k ≥ 0,

in the almost isospectral case.

1.2. Harmonic oscillator. Consider first the harmonic oscillator with zeroground-state energy:

(15) V (x) =ω2

4x2 − ω

2.

which is a case treated by Tkatchuk [19], Fellows and Smith [15]. Its spectrum iswell known:

(16) En = nω; ψn (x) ∼ Hn (ωx/2) exp(−ωx2/4

)and the corresponding RS functions wn(x) can be written as terminating continuedfractions [21]. We then obtain for its ”spatially Wick rotated” image vn(x) =−iwn(ix):

(17) vn(x) =ω

2x+

nω

ωx+ ... (n− j + 1)ω

ωx+ ... 1

x.


Clearly vn(x) does not present any singularity on the positive real half line.The recurrence between the RS functions gives:

(18) vn(x) = v0(x) +En

v0(x) + vn−1(x)

and vn(x) has the same odd parity as v0(x). Then vn(x) is regular on all R and its

asymptotic behaviour at ∞ is the one of v0(x). The normalizability of ψ(n)0 (x) ∼

exp(−∫vn(x)dx

)is then ensured.

V (x) satisfies Eq.(10) with δ = ω and Eq.(3) gives

(19) V (n)(x) = V (x) + (n+ 1)ω.

The spectrum of the corresponding hamiltonian H(n) is then:

(20) E(n)k = (k + n+ 1)ω; ψ

(n)k (x) = ψk (x) , k ≥ 0

and its SUSY partner H(n) has the following associated potential:

(21) V (n)(x) = 2v2n(x)−ω2

4x2 − (n+ 1)ω.

where vn(x) is the rational function given in Eq.(17). For each n, V (n)(x) consti-

tutes a regular rational extension of V (x) and the spectrum of H(n) is completely

determined. Indeed, since ψ(n)0 (x) is normalizable we are in the almost isospectral

case and Eq.(14) gives:(22){

E(n)0 = 0; ψ

(n)0 (x) ∼ exp

(−∫vn(x)dx

)E

(n)k+1 = (k + n+ 1)ω; ψ

(n)k+1 (x) = ((n+ k + 1)ω)−1/2

(− d

dx+ vn(x)

)ψk (x) , k ≥ 0.

We recover here the results obtained by Tkachuk [19], Fellows and Smith [15]and in a different way by Gomez-Ullate, Kamran and Milson [4]. In particular, for

n = 1 V (1)(x) is the l = 1 isotonic potential

(23) V (1)(x) =ω2

4x2 +

2

x2+

ω

2

and for n = 2, V (2)(x) is the CPRS [14] potential

(24) V (2)(x) =ω2

4x2 + 4ω

ωx2 − 1

(ωx2 + 1)2 +

3

2ω.

This last appears to be a particular exactly solvable case of the generalizedquantum isotonic potential, the spectrum of which has been recently studied in adetailed manner by Saad et al [22].

2. Second category potentials

2.1. Definition. As shown in [21], the translationally shape invariant poten-tials can be classified into two categories in which the potentials can be brought intoa harmonic or isotonic form respectively, using a change of variable which satisfiesa constant coefficient Riccati equation. The first element of the second categoryis the isotonic oscillator potential itself (ie the radial effective potential for a threedimensional isotropic harmonic oscillator with zero ground-state energy)

(25) V (x; l) =ω2

4x2 +

l(l + 1)

x2− ω

(l +

3

2

), x > 0.


Its spectrum is given by:

(26) En = 2nω, ψn (x; l) ∼ exp

(−∫

wn (x; l) dx

),

where the excited state Riccati-Schrodinger function (RS function) wn (x; l) can beexpressed as a terminating continued fraction as

(27)

wn(x; l) =ω

2x−

l + 1

x−

2nω

ωx − (2l + 3) /x−� ... � 2 (n − j + 1)ω

ωx − (2 (l + j) + 1) /x−� ... � 2ω

ωx − (2 (l + n) + 1) /x.

All the others second category potentials (with a zero ground-state energyE0 = 0) can be brought to the form

(28) V±(y±;λ, μ) = λ (λ∓ α) y2± +μ (μ− α)

y2±+ V0±(λ, μ),

with V0±(λ, μ) = −α (λ± μ)− 2λμ. The variable y± is defined via:

(29)dy±(x)

dx= α± αy2±(x),

that is, y+(x) = tan (αx+ ϕ0) ,in the V+ case (+ type) and y−(x) = tanh (αx+ ϕ0)or y− = coth (αx+ ϕ0) , in the V− case (− type). We call V+(y+;λ, μ) andV−(y−;λ, μ) dual partners (in the sense of Krajewska, Ushveridze and Walczak[20]) and we have

(30) V±(y;λ, μ) = V∓(y;λ1,∓, μ) + V0±(λ, μ)− V0∓(λ1,∓, μ),

with λ1,± = λ± α.The spectrum (En±, w±n) of H±(λ, μ) = −d2/dx2 + V±(y± (x) ;λ, μ) is known

analytically [21]. We have for the energies:

(31) En±(λ, μ) = ± (φ2,± (λn,±, μn)− φ2,± (λ, μ)) ,

with φ2,± (λ, μ) = (λ± μ)2 and (λn,±, μn) = (λ± nα, μ+ nα).As for the RS functions, they are given by:

wn,±(x;λ, μ) = λy±(x)−μ

y±(x)− En±(λ, μ)

(λ+ λ1,±) y±(x)− (μ+ μ1) /y±(x)− ...(32)

En±(λ, μ)− Ej−1±(λ, μ)

(λj−1,± + λj,±) y±(x)− (μj−1 + μj) /y±(x)− ...

En±(λ, μ)−En−1±(λ, μ)

(λn−1,± + λn,±) y±(x)− (μn−1 + μn) /y±(x)

and in particular:

(33) w0,±(x;λ, μ) = λy±(x)−μ

y±(x).

This category includes the Scarf and the Darboux-Poschl-Teller families of po-tentials [23, 24, 1, 2] and encompasses the second and third Gendenshtein ansatze[3]. Scarf I and Darboux-Poschl-Teller I are of + type the and Scarf II, Darboux-Poschl-Teller and Darboux-Poschl-Teller II are their respective− type dual partners[21].


2.2. Isotonic oscillator. The ”spatially Wick rotated” image of the RS func-tion Eq.(27) is for this model:

(34) vn(x; l) = v0(x; l) +Rn(x; l),

where

(35) v0(x; l) =ω

2x+

l + 1

x

and

(36) Rn(x; l) =2nω

ωx+ (2l+ 3) /x+ ... 2 (n− j + 1)ω

ωx+ (2 (l+ j) + 1) /x+ ... 2ω

ωx+ (2 (l+ n) + 1) /x.

Clearly vn(x, l) does not present any singularity on the positive real half line.It has to be noticed that the term (l + 1) /x which then appears in every vn(x),induces a nonnormalizable singularity at the origin for exp

(−∫vn(x; l)dx

). For

instance

(37) exp

(∓∫

v0(x; l)dx

)= x∓(l+1) exp

(∓ω

4x2),

are not square integrable and we are consequently in the case of a strict isospec-trality. The condition Eq.(10) is satisfied with δ = 2ω

(l + 3

2

). The ”Wick rotated”

potential is then given by

(38) V (n)(x; l) =ω2

4x2 +

l(l + 1)

x2+ 2

(n+ l +

3

2

)ω.

and the spectrum of H(n) (l) = −d2/dx2 + V (x; l) + 2(n+ l + 3

2

)ω is:

(39)

E(n)k (l) = 2

(k + n+ l +

3

2

)ω; ψ

(n)k (x; l) = ψk (x; l) ∼ exp

(−∫

wk(x; l)dx

), k ≥ 0.

Its SUSY partner H(n) has the following associated potential (see Eq.(11)):

(40) V (n)(x; l) = V (x; l)− 2v′0(x; l)− 2R′n(x; l) + 2

(n+ l +

3

2

)ω.

But we have(41)V (x; l)− 2v′0(x; l) = − (V (ξ; l) + 2w′

0(ξ, l))ξ=ix− δ = −V (ix; l+1)− δ = V (x; l+1)

and consequently

(42) V (n)(x; l) = V (x; l + 1)− 2R′n(x; l) + 2

(n+ l +

3

2

)ω,

where R′n(x; l) is a rational function regular on x > 0.

As in the harmonic oscillator case considered above, V (n)(x; l) constitutes a reg-ular rational extension of V (x; l) the spectrum of which is completely determined.Eq.(13) gives for k ≥ 0 :(43)

E(n)k (l) = 2

(n+ k + l+

3

2

)ω; ψ

(n)k (x; l) =

1√2(n+ k + l+ 3

2

)ω

(− d

dx+ vn(x; l)

)ψk (x; l) .

For instance, we have for n = 1

(44) V (1)(x; l) = V (x; l + 1) + 2

(l +

5

2

)ω +

4ω

ωx2 + 2l + 3− 8ω (2l + 3)

(ωx2 + 2l + 3)2


and we recover the first rationally-extended radial oscillator obtained by Quesne[8].

More generally, since in terms of Laguerre polynomials L(λ)n the isotonic oscil-

lator eigenstates take the form

(45) ψn (x; l) ∼ xl+1e−ωx2/4L(l+1/2)n

(ωx2/2

),

we have

(46) Rn(x; l) = log(L(l+1/2)n

(−ωx2/2

)).

In Odake-Sasaki ’s approach, it corresponds to a prepotential of the form

(47) Wn (x; l) = −ω

4x2 + (l + 1) log x+ log

(L(l+1/2)n

(−ωx2/2

))and we recover the result obtained in [13] for the potentials associated to the L1exceptional orthogonal polynomials.

Then we can still write(48)

V (n)(x; l) = V (x; l+1)+2

(n+ l +

3

2

)ω+

2ωx

L(l+1/2)n (−ωx2/2)

(∂L

(l+1/2)n (z)

∂z

)z=−ωx2/2

.

2.3. Other second category potentials. Consider a second category po-tential V±(x;λ, μ) as given in Eq(28). The RS function w±n(y;λ, μ) associated thelevel E±n(λ, μ) satisfies:

(49) −w′±n(x;λ, μ) + w2

±n(x;λ, μ) = V±(x;λ, μ)− E±n(λ, μ)

or introducing the variable defined in Eq(29):

(50) −α(1± y2±

)w′

±n(y±;λ, μ) + w2±n(y±;λ, μ) = V±(y±, λ, μ)− E±n(λ, μ).

If we set x → ix and y± (ix) = iu±(x), Eq(29) gives du±/dx = α ∓ αu2±. In

other words u±(x) = y∓(x).Define v±n(y∓;λ, μ) = −iwn∓(iy∓;λ, μ), that is

(51) v±n(x;λ, μ) = v±0(x;λ, μ) +R±n(x;λ, μ),

where

(52) v±0(x;λ, μ) = λy±(x) +μ

y±(x)

and

R±n(x;λ, μ) =E∓n(λ, μ)

(λ+ λ1∓) y± (x) + (μ+ μ1) /y± (x)+ ...(53)

En∓(λ, μ)− Ej−1∓(λ, μ)

(λj−1∓ + λj∓) y± (x) + (μj−1 + μj) /y± (x)+ ...

En∓(λ, μ)− En−1∓(λ, μ)

(λn−1∓ + λn∓) y± (x) + (μn−1 + μn) /y± (x),

which is regular when y± > 0.Eq(50) becomes:

(54) α(1∓ y2∓

)v′∓n(y∓;λ, μ) + v2∓n(y∓;λ, μ) = V

(n)∓ (y∓;λ, μ),


where (see Eq(28) and Eq(30)):

(55) V(n)± (y±;λ, μ) = En∓(λ, μ)− V∓(iy±;λ, μ) = V±(y±;λ1,±, μ) + C

(n)± (λ, μ),

with

(56) C(n)± (λ, μ) = En∓(λ, μ)− (V0∓(λ, μ) + V0±(λ1,±, μ)) .

We recover, up to a constant, a second category potential with a modified

multiparameter and the spectrum of h(n)± (λ, μ) = −d2/dx2+V

(n)± (y;λ, μ) is directly

deduced from the one of H±(λ1,±, μ):(57)

E(n)k± (λ, μ) = Ek±(λ1,±, μ) + C

(n)± (λ, μ); ψ

(n)k± (x;λ, μ) = ψk± (x;λ1,±, μ) , k ≥ 0.

Eq(54) can be written as:

(58) v′±n(x;λ, μ) + v2±n(x;λ, μ) = V(n)± (x;λ, μ)

and v±n(x;λ, μ) is the superpotential associated with V(n)± (λ, μ), the SUSY partner

of which being

(59)

V(n)± (x;λ, μ) = V

(n)± (x;λ, μ)−2v′

±n(x;λ, μ) = V±(x;λ1,±, μ)−2v′±0(x;λ, μ)−2R′

±n(x;λ, μ)+C(n)± (λ, μ).

But, using Eq(29), we have

(60) V±(x;λ1,±, μ)− 2v′±0(x;λ, μ) = V±(x;λ1,±, μ)− 2α(1± y2±

) dv±0(y±;λ, μ)

dy±,

that is, (see Eq(52))

(61) V(n)± (x;λ, μ) = V±(y±;λ, μ1)− 2R′

±n(x;λ, μ) + C(n)± (λ, μ).

The two hamiltonians h(n)± (λ, μ) and h

(n)± (λ, μ) are factorizable as:

(62)

{h(n)± = A

(n)+± A

(n)±

h(n)± = A

(n)± A

(n)+± ,

where A(n)± (λ, μ) = d/dx+ v±n(x;λ, μ) =

(α± αy2±

)d/dy± + v±n(y±;λ, μ).

But

exp

(−∫

v±0(x;λ, μ)dx

)∼ y

−μ/α±

(1± y2±

)(μ±λ)/2α, ψ

(n)k± (x, λ, μ) ∼ exp

(−∫

v±n(x;λ, μ)dx

)

and their inverses have non allowed singular behaviours. We are then again in thecase of a strict isospectrality

The spectrum of H(n)± (λ, μ) is:

(63) E(n)±,k(λ, μ) = En,±(λ1,±, μ), ψ

(n)±,k (x;λ, μ)

=1√

Ek,±(λ1,±, μ)(−d/dx+ v±n(x;λ, μ))ψ±,k (x, λ1,±, μ) , k ≥ 0.

We have then obtained a family of potentials(64)

Vn±(y±;λ, μ) = V(n)± (x;λ, μ)− C

(n)± (λ, μ) = V±(y±;λ, μ1)− 2α

(1± y2

±) dR±n(y±;λ, μ)

dy±,

strictly isospectral to V±(x;λ1,±, μ). Each member of this family constitutes asolvable regular extension of V± which is rational in the y± variable.


In the particular case n = 1, Eq(64) gives(65)

V1±(y±;λ, μ) = V±(x;λ, μ1) + 8α2 (λ∓ μ∓ α)

(1± y2

±) (

(2λ∓ α) y2± (x)− (2μ+ α)

)((2λ∓ α) y2

± (x) + (2μ+ α))2 .

To illustrate this result, consider the example of the Darboux-Poschl-Teller I po-tential with zero-energy ground state which is given by [1, 2, 21]:

(66)

VPT1(x;λ, μ) = λ (λ − α) y2+(x) +

μ (μ − α)

y2+(x)

+ V0+(λ, μ) = −(λ+ μ)2+

λ(λ − α)

cos2 (αx)+

μ(μ − α)

sin2 (αx), x ∈

]0,

π

2α

[,

where y+(x) = tan (αx) with V0+(λ, μ) = −α(λ+ μ)− 2λμ. Its energy spectrum is

En,PT1(λ, μ) = 4α2 (n− (λ+ μ) /2α)2 − (λ+ μ)2.Note that the Scarf I potential is directly obtained from the preceding one

by making the substitution (λ, μ) → ((λ+ μ) /2, (λ− μ) /2), α → α/2 in Eq(66)which gives

VS1(x;λ, μ) = λ (λ− α/2) y2+(x) +

μ (μ− α/2)

y2+(x)

− α

2(λ+ μ)− 2λμ(67)

= −λ2 +

(λ2 + μ2 − λα

)sin2 (αx)

− μ(2λ− α)cot (αx)

sin (αx), x ∈

]0,

π

α

[,

where y+(x) = tan (αx/2).The dual partner of the Darboux-Poschl-Teller I potential is the Darboux-

Poschl-Teller II potential

VPT2(x;λ, μ) = (λ− μ)2 − λ(λ+ α)

cosh2 (αx)+

μ(μ− α)

sinh2 (αx)(68)

= λ (λ+ α) y2−(x) +

μ (μ− α)

y2−(x)

+ V0−(a), B < A , x > 0,

where y(x) = tanh (αx) with V0−(λ, μ) = −α(λ − μ) − 2λμ. Its energy spectrum

is given by En,PT2(λ, μ) = (λ− μ)2 − 4α2 (n− (λ− μ) /2α)

2.

From Eq(65) we deduce for n = 1, that VPT1(x;λ+ α, μ) and

V1,PT1(x;λ, μ)(69)

= VPT1(x;λ, μ + α) + 8α2(λ − μ − α)

(μ + α − λ) − (λ + μ) cos (2αx)

((λ + μ) + (μ + α − λ) cos (2αx))2

= VPT1(x;λ, μ + α) + 8α2

(λ + μ

(λ + μ) + (μ + α − λ) cos (2αx)−

(2μ + α) (2λ − α)

((λ + μ) + (μ + α − λ) cos (2αx))2

)

are strictly isospectral.Up to a shift on the λ parameter(λ → λ + α), this result is similar to the one

obtained by Odake and Sasaki [11].

3. Conclusion

We have shown that every translationally shape invariant potential of secondcategory admits a family of solvable regular extensions which are rational in anappropriate variable. All the members of this family are isospectral to the orig-inal potential and the associated eigenstates are easily related to the initial onesby application of first order differential operators. The eigenfunctions of such ra-tional extensions contain exceptional (Xl) Laguerre and Jacobi polynomials whichare the subject of recent active research [8, 9, 11, 4, 5, 25, 26, 12, 13]. Theapproach developed to build these families is based on a generalisation of the usualSUSY QM approach using ”Wick rotated” higher excited states as superpotentials.


It rests on purely algebraic manipulations which are direct and transparent andoffers a new perspective on the structure of the extended potentials. A differentpoint of view has been recently proposed by the author in which a more generalregularization procedure is developed on the basis of the specific symmetries ofthe initial potential, the ”Wick rotation” appearing as equivalent to one of thesesymmetries [27]. Starting from the isotonic system this allows to generate in asystematic way all the rational extensions of the L1, L2 and L3 series. This schemecan be applied to the shape invariant potentials of the first category [28] givingin particular interesting new results about the possible rational extensions of theeffective radial Kepler-Coulomb potential. It admits also a n-step generalization[29, 30], the eigenfunctions of the obtained extensions containing in the n = 2 casethe new exceptional Laguerre polynomials recently discovered by Gomez-Ullate etal [31, 32].

Acknowledgments

We would like to thank Professor P.G.L. Leach for useful suggestions and acareful reading of the manuscript.

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337-347 (2009).[10] B. Bagchi and C. Quesne, “An update on PT -symmetric complexified Scarf II potential,

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[16] M. Robnik, “ Supersymmetric quantum mechanics based on higher excited states”, J. Phys.A 30, 1287-1294 (1997). MR1449282 (98d:81022)

[17] R. Klippert and H. C. Rosu, “ Strictly isospectral potentials from excited quantum states”,Int. J. Theor. Phys., 4, 331-340 (2002). MR1888444

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spectra of anharmonic oscillators”, Chaos 4, 47 -53 (1994). MR1268969 (95a:81055)[19] V. M. Tkachuk, “ Supersymmetric method for constructing quasi-exactly and conditionally-

exactly solvable potentials”, J. Phys. A 32, 1291-12 (1999). MR1690676 (2000d:81047)[20] A. Krajewska, A. Ushveridze and Z. Walczak, “ On the duality of Quasi-ExactlySolvable

problems”, Arxiv preprint hep-th/9601025 (1996).[21] Y. Grandati and A. Berard, “Rational solutions for the Riccati-Schrodinger equations as-

sociated to translationally shape invariant potentials”, Ann. Phys. 325, 1235-1259 (2010).MR2657537 (2011e:81108)

[22] N. Saad, R. L. Hall, H. Ciftci and O. Yesiltas, “Study of the generalized quantum isotonicnonlinear oscillator potential”, Adv. Math. Phys. 2011, 750168 (2011). MR2801350

[23] G. Darboux, Lecons sur la theorie des surfaces Vol 2, 2nd ed., 210-215 (Gauthier-Villars,Paris, 1915).

[24] G. Poschl and E. Teller, “Bemerkungen zur Quantenmechanik des anharmonischen Oszilla-tors”, Z. Phys. 83, 143-151 (1933).

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arXiv:1108.4503 (2011). MR2812879[30] S. Odake and R. Sasaki, “ Exactly Solvable Quantum Mechanics and Infinite Families of

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Institut de Physique, Equipe BioPhyStat, ICPMB, IF CNRS 2843, Universite Paul

Verlaine-Metz, 1 Bd Arago, 57078 Metz, Cedex 3, France

Institut de Physique, Equipe BioPhyStat, ICPMB, IF CNRS 2843, Universite Paul

Verlaine-Metz, 1 Bd Arago, 57078 Metz, Cedex 3, France


The pentagram map: geometry, algebra, integrability

Valentin Ovsienko

Abstract. We discuss recent results and open problems related to a veryspecial discrete dynamical system called the pentagram map. The pentagrammap acts on the moduli space Cn of projective equivalence classes of n-gons inthe projective plane. Its continuous limit is the famous Boussinesq equation.The most remarkable property of the pentagram map is its complete integra-bility recently proved for the (larger) space of twisted n-gons. Integrability ofthe pentagram map on Cn is still an open problem. We discuss the relation ofthe pentagram map to the space of 2-frieze patterns generalizing that of theclassical Coxeter-Conway frieze patterns. This space has a structure of clustermanifold. It also has a natural symplectic form.

Contents

1. Introduction: The pentagram map2. Complete integrability of the pentagram map on the space Pn

3. Relation to the Boussinesq equation4. The space Cn and 2-frieze patternsAcknowledgmentsReferences

1. Introduction: The pentagram map

Integrability is one of the central notions in mathematics. Starting from Eulerand Jacobi, the theory of integrable systems is among the most remarkable applica-tions of geometric ideas to mathematics and physics. Discrete integrable systems isa new and actively developing subject. However, geometric interpretation of mostof the discrete integrable systems considered in the mathematical and physical lit-erature is unclear. In this short survey, I will present recent results concerningone particular discrete integrable system that naturally appear in the context ofprojective differential geometry.

The pentagram map T is a dynamical system introduced by Richard Schwartzin 1992 [9] and studied in a series of recent papers [10, 11, 7, 4]. The map Twas initially defined on the space of (convex) closed n-gons in RP2. Given such an

2010 Mathematics Subject Classification. Primary 37J35; Secondary 51A99.Key words and phrases. Pentagram map, Cluster algebra, Frieze pattern, Moduli space.


127

128 VALENTIN OVSIENKO

n-gon P , the corresponding n-gon T (P ) is the convex hull of the intersection pointsof consecutive shortest diagonals of P , see Figure 1.

T(P)T(P)

P P

Figure 1. Examples of the pentagram map

The pentagram map commutes with projective transformations. That is,φ(T (P )) = T (φ(P )), for any φ ∈ PGL(3,R). We denote the space of all n-gonsmodulo projective equivalence by Cn:Cn =

{(v1, . . . , vn) ∈ P2 | det(vi, vi+1, vi+2) �= 0, i = 1, . . . , n

}/PGL(3,R),

where the vertices vi are cyclically numerated by i ∈ Zn. The space Cn is a (2n−8)-dimensional algebraic variety. The pentagram map is then defined as a map from Cto itself. Note that the space C is a higher analog of the moduli spaceM0,n of genuszero curves with n marked points that plays an important role in mathematics.

Let us also consider a slightly larger space that contains Cn. A twisted n-gonis a map v : Z→ RP2 such that

vi+n = M ◦ vi, i ∈ Z,

for some fixed element M ∈ PGL(3,R) called the monodromy. We assume thatvi−1, vi, vi+1 are in general position for all i. We denote by Pn the space of twistedn-gons modulo projective equivalence. The space of twisted n-gons is much easierto study than the space Cn. This space is naturally isomorphic to two differentspaces, namely, the space of third-order difference equations, see Section 2.1 andthe space of 2-frieze patterns, see Section 4.

One thus obtains two versions of the pentagram map:

T : Cn → Cn, and T : Pn → Pn.

Of course, the first one is a restriction of the second.Integrability of T on Cn (in the classical sense of Arnold–Liouville) was exper-

imentally checked and then conjectured by Schwartz [9, 10, 11]. This conjectureturned out to be difficult. The first result was obtained in [7], where it was shownthat T is completely integrable on the space of twisted n-gons Pn. The work onthe initial conjecture is still in progress [8]. Let us also mention beautiful geometricproperties of the pentagram map [12, 13].

2. Complete integrability of the pentagram map on the space Pn

The main result of [7] is the following theorem esbablishing complete integra-bility of the map T on the space of twisted n-gons.

PENTAGRAM MAP 129

Theorem 1. The pentagram map T : Pn → Pn is completely integrable in theLiouville-Arnold sense.

This means, there exists a T -invariant Poisson structure on Pn and “sufficientlymany” Poisson-commuting T -invariant functions that (generically) span the nullspace of the Poisson structure.

2.1. Difference equations. It is a powerful general idea of projective differ-ential geometry to represent geometrical objects in an algebraic or analytic way.

Consider a difference equation of the form

(2.1) Vi = ai Vi−1 − bi Vi−2 + Vi−3,

where ai, bi ∈ C or R are n-periodic: ai+n = ai and bi+n = bi, for all i. A solutionV = (Vi) is a sequence of numbers Vi ∈ C or R satisfying (2.1). Since the spaceof solutions of (2.1) is 3-dimensional, we often understand Vi as vectors in R3.The n-periodicity then implies that there exists a matrix M ∈ SL(3,R) called themonodromy matrix, such that Vi+n = M Vi. Note that the monodromy matrix Mbelongs to SL(3,R) (i.e., detM = 1) because the coefficient at Vi+3 is unit.

The following result was proved in [7].

Proposition 2.1. If n is not divisible by 3, i.e., if n = 3m+1 or n = 3m+2,then:

(i) the space Pn is isomorphic to the space of the equations ( 2.1);(ii) the subspace Cn ⊂ Pn is isomorphic to the space of the equations ( 2.1) with

trivial monodromy, M = Id.

It follows that the functions (ai, bi) are (globally defined) coordinates on Pn.One thus obtains the following identifications:

Pn∼= R2n, Pn

∼= C2n,

in the real (resp. complex) case, provided n �= 3m. The condition M = Id is a(very complicated) algebraic identity. Therefore, the space Cn ⊂ R2n (resp. C2n)is an algebraic variety (of codimension 8).

It worth noting that the space of difference equations (2.1) is one of the mostclassical space used in the theory of discrete integrable systems, see [3].

2.2. Explicit formula for T . It is not very difficult to describe the pentagrammap in terms of the coordinates (ai, bi) defined via the above identification of thespace Pn with the space of difference equations (2.1).

Denote by T ∗(ai) and T ∗(bi) the pull-backs of the coordinates by the map T .

Proposition 2.2. Assume that n = 3m+ 1 or n = 3m+ 2; in both cases,(2.2)

T ∗(ai) = ai+2

m∏k=1

1 + ai+3k+2 bi+3k+1

1 + ai−3k+2 bi−3k+1, T ∗(bi) = bi−1

m∏k=1

1 + ai−3k bi−3k−1

1 + ai+3k bi+3k−1.

This statement was obtained in [7]1.

1Formula for T ∗(bi) in [7] contains a typo.


2.3. The Poisson bracket. The first ingredient of the complete integrabilityresult is a T -invariant Poisson bracket on the space Pn.

Assume, as above, that n = 3m + 1 or n = 3m + 2. In both cases, we definethe following Poisson bracket.

(2.3)

{ai, aj} =m∑

k=1

(δi,j+3k − δi,j−3k) ai aj ,

{ai, bj} = 0,

{bi, bj} =

m∑k=1

(δi,j−3k − δi,j+3k) bi bj .

Proposition 2.3. The Poisson bracket ( 2.3) is invariant with respect to thepentagram map.

This statement can be proved using the explicit formula (2.2), see [7] for thedetails. An easy analysis shows:

Proposition 2.4. The Poisson bracket ( 2.3) is of co-rank 2 when n is odd andco-rank 4 when n is even.

Indeed, two Casimir functions are given by the products of the coordinate functions:

A = a1 · · · an, B = b1 · · · bn.

Furthermore, if n is even, then one gets two more Casimirs:

A′ = a2 a4 · · · an, B′ = b2 b4 · · · bn.

2.4. The spectral parameter and the monodromy invariants. In thissection we describe the invariants of the pentagram map. These invariant functionswere found in [11] (see also [7]) and called the monodromy invariants . It was thenproved in [7] that these functions Poisson commute.

The coefficients of the monodromy operator M ∈ SL3(R) are functions on Pn

polynomial in the coordinates (ai, bi). The trace of the monodromy, tr(M), isa T -invariant polynomial function on Pn. In order to construct a big family ofinvariants, we will use the following rescaling parameter.

Consider the one-parameter group of diffeomorphisms ϕt : Pn → Pn

ϕt :(ai → t ai, bi → 1

t bi).

The explicit formula (2.2) implies:

Proposition 2.5. The pentagram map T commutes with the diffeomorphismsϕt.

We thus obtain the following decomposition

tr(M) =∑

Ik,

where Ik are homogeneous in t components of the polynomial tr(M). The func-tions Ik are also invariants of the pentagram map. The explicit formulas for themonodromy invariants can be found in [11, 7].

The following result is obtained in [7].

PENTAGRAM MAP 131

Proposition 2.6. The monodromy invariants Ik Poisson commute, that is

{Ik, I�} = 0,

for all k, �.

The functions In are all nontrivial polynomials. It is shown in [11] that thereare exactly 2[n/2] + 2 functions In and that they are algebraically independentalmost everywhere on Pn.

This completes the proof of Theorem 1.

3. Relation to the Boussinesq equation

The theory of infinite-dimensional integrable systems on functional spaces ismuch more developed than the theory of discrete integrable systems. Much ofthe information about the pentagram map can be obtained by investigating then → ∞ “continuous limit”. It turns out that the continuous limit of T is theclassical Boussinesq equation. In particular, the Poisson bracket (2.3) is a discreteversion of the well-known first Poisson structure of the Boussinesq equation.

3.1. Non-degenerate curves and differential operators. The continuouslimit of a twisted n-gon is a smooth parametrized non-degenerate curve γ : R→ RP2

with monodromy:

(3.1) γ(x+ 1) = M(γ(x)),

for all x ∈ R, where M ∈ PGL(3,R) is fixed. The vectors γ′(x) and γ′′(x) arelinearly independent for all x ∈ R. As in the discrete case, we consider classes ofprojectively equivalent curves. The continuous analog of the space Pn, is the spaceof parametrized non-degenerate curves in RP2 up to projective transformations.This space is very well known in classical projective differential geometry, see, [6]and references therein.

There exists a one-to-one correspondence between classes of projectively equiv-alent curves and linear differential operators

(3.2) A =

(d

dx

)3

+1

2

(u(x)

d

dx+

d

dxu(x)

)+ w(x)

where u and w are smooth periodic functions.

3.2. Continuous limit of the pentagram map. Let us now define a con-tinuous analog of the map T . Given a non-degenerate curve γ(x), at each point xwe draw a small chord: (γ(x− ε), γ(x+ ε)) and obtain a new curve, γε(x), as theenvelop of these chords, see Figure 2.

(x) (x+ )

(x- )(x)

Figure 2. Evolution of a non-degenerate curve


The differential operator (3.2) corresponding to γε(x) contains new functions(uε, wε). It is shown in [7] that

uε = u+ ε2u+O(ε3), wε = w + ε2w +O(ε3)

with some explicit u, w. We then assume that the functions u(x) and w(x) dependon an additional parameter t (the “time”) and define an evolution equation:

u = u, w = w

that we understand as a vector field on the space of functions (and therefore ofoperators (3.2)). Here and below u and w are the partial derivatives in t, thepartial derivatives in x will be denoted by ′.

The second main result of [7] is as follows.

Theorem 2. The continuous limit of the pentagram map T is the followingequation:

(3.3)

u = w′,

w = −uu′

3− u′′′

12.

This is the classical Boussinesq equation.

3.3. The Poisson structure. The equation (3.3) is bi-Hamiltonian. We de-scribe here the simplest Poisson structure called the first Poisson structure of theBoussinesq equation.

Consider the space of functionals of the form

H(u,w) =

∫S1

h(u, u′, . . . , w, w′, . . .) dx,

where h is a polynomial. The variational derivatives are the smooth functions onS1 given by

δuH =∂h

∂u−(∂h

∂u′

)′+

(∂h

∂u′′

)′′−+ · · ·

The constant Poisson structure on the space of functionals is defined by

(3.4) {G,H} =∫S1

(δuG (δwH)

′+ δwG (δuH)

′)dx.

Given a functional H, the corresponding Hamiltonian vector field is given by

u = (δwH)′,

w = (δuH)′ .

In particular, the function

H =

∫S1

(w2

2− u3

18− uu′′

24

)dx

is the Hamiltonian function for the equation (3.3).The Poisson bracket (2.3) was obtained in [7] by discretization of the bracket

(3.4).

PENTAGRAM MAP 133

4. The space Cn and 2-frieze patterns

Unlike the space of twisted polygons Pn, the space of closed polygons has avery nontrivial geometry (and topology). In this section, we explain a beautifulcombinatorics associated to the space Cn.

4.1. The definition of 2-frieze patterns. A 2-frieze pattern is a grid ofnumbers, or polynomials, rational functions, etc., (vi,j)(i,j)∈Z2 and (vi+ 1

2 ,j+12)(i,j)∈Z2

organized as follows

��

��

�� vi− 12 ,j+

12

��

��

��

�

��

��

��

. . .

��

��

vi− 12 ,j−

12

��

��

�

��

vi,j

��

��

��

��

vi+ 12 ,j+

12

��

��

�

��

��

vi− 12 ,j−

32

��

��

�vi,j−1

��

��

��

��

vi+ 12 ,j−

12

��

��

��

�vi+1,j

��

��

��

��

· · ·

��

vi,j−2

��

��

vi+ 12 ,j−

32

��

��

�

� � � � � �vi+1,j−1

��

��

��

��

vi+ 32 ,j−

12

��

��

��

�vi+2,j

��

��

vi+1,j−2

��

��

vi+ 32 ,j−

32

� � � � � �

��vi+2,j−1

��

��

vi+2,j−2

such that every entry is equal to the determinant of the 2× 2-matrix formed by itsfour neighbours:

B

��

��

��

F

��

��

A

��

�� E

��

��

��

��

D

��

��

� H �� E = AD −BC, D = EH − FG, . . .

C G

Generically, two consecutive rows in a 2-frieze pattern determine the whole 2-frieze pattern. The notion of 2-frieze pattern is an analog of the classical notion ofCoxeter-Conway frieze pattern [1, 2].

A 2-frieze pattern is called closed if it is bounded from above and from belowby two rows of 1’s:

· · · 1 1 1 1 1 · · ·· · · v0,0 v 1

2 ,12

v1,1 v 32 ,

32

v2,2 · · ·...

......

......

· · · 1 1 1 1 1 · · ·We call the width of a closed pattern the number of the rows between the two rowsof 1’s.


4.2. Relation to the space Cn. Let us assume that n �= 3m. In order toobtain a relation between difference equations and 2-friezes, we assume: vi,i =ai, vi− 1

2 ,i−12= bi and consider the 2-frieze bounded from above by a row of 1’s:

(4.1)

· · · 1 1 1 1 1 · · ·· · · b1 a1 b2 a2 b3 · · ·

· · · b1b2 − a1 a1a2 − b2 b2b3 − a2 · · ·...

......

Note that the coefficients in the above 2-frieze are periodic: ai+n = ai and bi+n = bi.We will call such a 2-frieze pattern 2n-periodic.

The following statement is one of the main results of [5].

Theorem 3. A 2n-periodic 2-frieze pattern (4.1) is closed if and only if themonodromy of the corresponding difference equation (2.1) is trivial: M = Id.

One thus obtains three equivalent ways to understand the space Cn, providedn �= 3m:

(1) as the space of modules of closed n-gons in P2;(2) as the space of difference equations (2.1) with trivial monodromy;(3) as the space of closed 2n-periodic 2-frieze patters.

A number of algebraic and geometric results are obtained in [5] as corollary ofthe above theorem. In particular, it is proved that the space Cn has a structure ofcluster manifold. It is also shown that this space has a symplectic structure. Wehope that these results will help to understand the geometry of the pentagram mapon the space Cn.

Acknowledgments

The author is grateful to Sophie Morier-Genoud, Sergei Tabachnikov andRichard Schwartz for helpful discussions. The author also thanks the referees for anumber of helpful comments.

References

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(1973), 87–94 and 175–183. MR0461269 (57:1254)[3] E. Frenkel, N. Reshetikhin, M. Semenov-Tian-Shansky, Drinfeld-Sokolov reduction for dif-

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arXiv:1004.4311.

CNRS, Institut Camille Jordan, Universite Claude Bernard Lyon 1, 43 boulevard

du 11 novembre 1918, 69622 Villeurbanne cedex, France


Jet Bundles, symmetries, Darboux transforms

Enrique G. Reyes

Abstract. We review some aspects of the geometric approach to differentialequations due to mathematicians such as Sophus Lie, Gaston Darboux andElie Cartan. We consider some basic aspects of the formulation of differentialequations using jet bundles and, as a non-trivial example, we state Gromov’sh-principle and apply it to systems of differential equations: following somerecent work on holonomic approximations due to Eliashberg and Mishachev,

we present a geometric theorem on local existence of approximate solutionsto PDEs. We then review the theory of symmetries of differential equationswith particular emphasis on internal and nonlocal symmetries. We advance avery natural approach to nonlocal symmetries using exterior differential sys-tems, and we argue, motivated by calculations carried out in the context ofthe Camassa-Holm equation, that nonlocal symmetries can be considered asgeneralizations of the internal symmetries introduced by E. Cartan. Finally weexplain, using the associated Camassa-Holm equation, how to derive Darbouxtransformations using nonlocal symmetries and pseudo-potentials.

Contents

1. Introduction2. Jet bundles and differential equations3. A glimpse of Gromov’s h-principle4. Symmetries of differential equations5. Darboux transformations for the ACH equationReferences

1. Introduction

Partial differential equations (PDEs) first appeared as essential tools for theanalytic study of physical models; posteriorly, as explained for instance in [7], theybecame fundamental for mathematics itself. Of the very many deep instances ofinteraction between PDEs and other areas of mathematics we mention, as just oneexample, U. Pinkall and I. Sterling’s proof of the existence of an infinite number

2010 Mathematics Subject Classification. Primary 37K10, 70H06; Secondary 70G65, 70S10,53C23.

Key words and phrases. Jet Bundles, h-principle, internal symmetries, nonlocal symmetries,pseudo-potentials, Camassa-Holm equation, Darboux transforms.

Research partially supported by Fondo Nacional de Desarrollo Cientıfico y Tecnologico(FONDECYT) grants #1070191 and #1111042.


137

138 ENRIQUE G. REYES

of constant mean curvature tori [41], carried out by carefully analyzing the sinh–Gordon equation

uxx + utt + 2 sinh 2u = 0

and its generalized symmetries. On the other hand, other fields of mathematicshave played important roles in the understanding of differential equations as math-ematical objects of their own. Functional analysis and topology come immediatelyto mind, see [7, 21], but we can also study differential equations profitably bygeometric means.

The geometric approach to differential equations goes back to the fundamentalcontributions of Sophus Lie on the symmetry theory for general systems of differen-tial equations [32, 36] and owes much of its earlier development to the research ondifferential geometry itself, as the impressive treatise on the geometry of surfacesby Gaston Darboux [15] testifies. During the first half of the XXth Century, part ofthis research evolved into the modern theory of exterior differential systems in thehands of E. Cartan, see for example his classic book [11] and the modern treatise[8], while another part, the one more directly related to Lie’s work, became a welldeveloped and far reaching formal geometric theory of differential equations withmanifold applications [5, 10, 32, 36].

This paper presents some basic aspects of the rich interplay among topics ap-pearing in the research of Lie, Darboux and Cartan. Specifically, we consider Liesymmetries and their (local/nonlocal) generalizations [5, 32, 36], we present brieflyCartan’s internal symmetries (see [5]), we explain why nonlocal symmetries can beconsidered as a general kind of internal symmetries, and we construct Darbouxtransformations. The precise contents of this work are as follows:

Section 2 is on the differential geometric framework used in the study of dif-ferential equations. Most of the material in this section is well-known to specialists,but it may be of interest to the general reader, and it is needed for our discussionon symmetries. In Section 3 we present an all-too-brief account of Gromov’s h-principle, as a direct application of the structures introduced in Section 2. Weinclude a short discussion on holonomic approximation, following Eliashberg andMishachev [20], and we state one of its applications: the construction of approxi-mated solutions to PDEs. In Section 4 we consider local and nonlocal symmetriesof differential equations. The local theory is of course classical [5, 10, 32, 36],but there is still room for research in the nonlocal domain. We present a re-cent approach to nonlocal symmetries [45, 44, 25, 26] which is in turn based onthe theory of coverings and diffieties of J. Krasil’shchik, A. Vinogradov and theircoworkers [49, 30, 31, 32], and we advance a possible generalization of the the-ory by means of exterior differential systems. As a computational application, werecall a recent classification of nonlocal symmetries of the Camassa-Holm and as-sociated Camassa-Holm equations after [25, 26]. Finally, in Section 5, we show bymeans of an example how we can construct Darboux transformations using nonlocalsymmetries.

2. Jet bundles and differential equations

First of all, we observe that a differential equation (hereafter understood aseither a scalar equation or a system) Ξ = 0, in which Ξ depends on independentvariables x1, . . . , xn, dependent variables u

1, . . . , um, and a finite number of partialderivatives of the functions uα with respect to the variables xj , can be thought

JET BUNDLES, SYMMETRIES, DARBOUX TRANSFORMS 139

of as determining a locus in a suitable space in which x1, . . . , xn, u1, . . . , um, and

the partial derivatives ∂kuα/∂xkj are coordinates. By restricting this locus, we can

assume that the equation Ξ = 0 determines a submanifold of this space called the“equation manifold” of Ξ = 0. We now present some basic constructions whichformalize these comments. Our main tool is Charles Ehresmann’s theory of jets[18, 19] and its generalization to infinite dimensions [32].

2.1. Jet bundles. We begin with an (n + m)–dimensional fiber bundle π :E → M . The n-dimensional manifold M is the space of independent variablesxi, 1 ≤ i ≤ n, and the m-dimensional typical fiber is the space of the dependentvariables uα, 1 ≤ α ≤ m. If the reader wishes to assume that the bundle π issimply Rn ×Rm → Rn, he/she can safely do so: one of the most interesting char-acteristics of the theory is that even if it looks like the researcher is “trivializing”everything because he/she is working locally, the fact that we are primarily inter-ested in properties of differential equations and their solutions makes the geometryhighly nontrivial. Perhaps the best examples of this are the ideal of contact formswhich we will introduce momentarily, and the theory of conservation laws as itappears in [4, 32, 36].

Let s1(xi) = (xi, sα1 (xi)) and s2(xi) = (xi, s

α2 (xi)) be two local sections of the

bundle E →M defined about a point p = (xi) in M . We say that s1 and s2 agreeto order k at p ∈ M if sα1 , s

α2 and all the partial derivatives of the functions sα1

and sα2 , up to order k, agree at p. This notion determines a coordinate-independentequivalence relation on local sections of E. We let jk(s)(p) represent the equivalenceclass of the section s at p, and we call this equivalence class the k–jet of s at p. Wehave

Definition 2.1. The k–order jet bundle of E is the space

JkE =⋃p∈M

Jk(p),

in which Jk(p) denotes the set of all the k–jets jk(s)(p) of local sections s at p.

The jet bundle JkE possesses a natural manifold structure; it fibers over J lE(l < k) and also overM : local coordinates on JkE are (xi, u

α0 , u

αi1, uα

i1i2, . . . , uα

i1...ik),

in which(2.1)

uα0 (j

k(s)(p)) = sα(p), uαi (j

k(s)(p)) =∂sα

∂xi(p), uα

i1i2(jk(s)(p)) =

∂2sα

∂xi1∂xi2

(p) ,

and so forth, where jk(s)(p) ∈ JkE and (xi) → (xi, sα(xi)) is any local sectionin the equivalence class jk(s)(p). In these coordinates, the projection maps πk

M :JkE → M and πk

l : JkE → J lE, l < k, are obvious. For instance, πkM is simply

(xi, uα0 , u

αi1, . . . , uα

i1...ik) → (xi).

Any local section s : (xi) → (xi, sα(xi)) of E lifts to a unique local section

jk(s) of JkE called the kth prolongation of s. In coordinates, jk(s) is the section

jk(s)(xi) =

(xi, s

α(xi),∂sα

∂xi1

(xi) , ...,∂ksα

∂xi1 . . . ∂xik

(xi) , ...

).

We use prolongations to define a special class of differential forms:

Definition 2.2. A differential form ω on JkE is a contact form if jk(s)∗ω = 0for all local sections s of E.


A careful discussion of contact forms appears in [37]. The set of contact forms,Ik(E), is an ideal of the ring of differential forms on JkE. Locally, Ik(E) isgenerated by the basic contact one–forms

(2.2) θαi1...ir = d uαi1...ir −

n∑j=1

uαi1...irj d xj , 0 ≤ r ≤ k − 1 .

Contact forms allow us to define the Cartan connection on JkE by specifyinghorizontal subspaces: a vector field X on JkE is horizontal if and only if iXω = 0for all one–forms ω ∈ Ik(E). It follows from the Frobenius theorem that thedistribution on JkE determined by the Cartan connection is not integrable. Indeed,it is not hard to check that the exterior derivative of θαi1...ir is given by

(2.3) dθαi1...ir =n∑

j=1

dxj ∧ θαi1...irj .

Locally, horizontal vector fields are linear combinations of the total derivative op-erators (here, and whenever convenient hereafter, we use the Einstein summationconvention)

(2.4) Dj =∂

∂xj+ uα

j

∂

∂uα+ uα

i1j

∂

∂uαi1

+ uαi1i2j

∂

∂uαi1i2

+ · · · .

Now we consider lifts using the Cartan connection. If V is a vector field on thebase space M , its kth prolongation, prkV , is defined as follows, see [30]: the actionof prkV on a smooth function f on JkE at a point σ = jk(s)(p) ∈ JkE, p ∈M , isgiven by

(2.5) prkV (σ) · f = V (p) · (f ◦ jk(s)) .

For instance, the kth prolongation of the partial derivative ∂/∂xj is precisely theoperator Dj defined in (2.4). On the other hand, a vector field V on E can be

canonically prolonged to an horizontal vector field pr(k)V on JkE [5, 30, 32, 36]:if

V =

n∑i=1

ξi(xj , uβ)

∂

∂xi+

m∑α=1

ϕα(xj , uβ)

∂

∂uα,

the kth prolongation of V is the vector field pr(k)V given by the formula

(2.6) pr(k)V =

n∑i=1

ξi∂

∂xi+

m∑α=1

∑0≤#J≤k

ϕαJ

∂

∂uαJ

,

in which J is a multi-index understood as either an unordered k-tuple of integersJ = (j1, . . . , jk), 1 ≤ j1, . . . , jk ≤ n or J = 0, #J = k, #0 = 0, and the functionsϕαJ are obtained inductively by means of

(2.7) ϕα0 = ϕα ; ϕα

J i = Di ϕαJ −

n∑k=1

Di(ξk) uα

J k .

We note that if V is a vector field on E, the flow Φε of V can be canonically

prolonged to a flow Φ(k)ε on JkE, see [36, p. 98]. The vector field pr(k)V just

defined is precisely the infinitesimal generator of Φ(k)ε , see [36, pp. 101, 110].


Definition 2.3. A smooth map Φ : JkE → JkE′ is called a contact transfor-mation if it preserves contact ideals, that is, if

(2.8) Φ∗Ik(E′) ⊆ Ik(E) .

Contact transformations are the most general kind of transformations betweenjet bundles which “send solutions to solutions”. Thinking in terms of coordinates,contact transformations are determined by mappings which mix independent vari-ables, dependent variables and derivatives of the dependent variables. Thus, thestandard distinction between independent and dependent variables is not preservedby contact transformations. On the other hand, contact transformations are not asgeneral as we would think. We quote a classical theorem by A.V. Backlund [5, 37]:

Theorem 2.4. Let Φ : JkE → JkE be a contact transformation. If the fibers ofE are one-dimensional, then Φ is the (k− 1)th prolongation of a first order contacttransformation, while if the fibers of E are of dimension greater than one, then Φis the kth prolongation of a transformation φ : E → E.

Example 2.5. An interesting contact transformation has been discovered byO. Morozov [35]: consider the generalized Hunter-Saxton equation

(2.9) uxt = uuxx + κu2x, k �= 1,

and the linear Euler-Poisson equation

(2.10) uxt =1

κ(t+ x)ut +

2(1− κ)

κ(t+ x)ux −

2(1− κ)

(κ(t+ x))2u .

Morozov proved in [35] that the contact transformation

u = (t+ x)−1/κ (κ(t+ x)ux + (κ− 1)u)

t = κ−1t

x = −(t+ x)(κ−1)/κ (κ(t+ x)ux − u)

ut = κ2(t+ x)−1/κ (ut − ux)

ux = −(t+ x)−1

maps the linear equation (2.10) into Equation (2.9) written in variables with tildes.

Further information on these important transformations appears for examplein [4, 5, 23, 32, 37]. For instance, it is shown in [23], following earlier work by R.Gardner (the precise reference appears in [23]) that the classical division of secondorder scalar PDEs in two independent variables into equations of hyperbolic, par-abolic and elliptic type can be made intrinsically, in a manifestly contact invariantfashion.

2.2. Infinite jet bundles. The infinite jet bundle π∞M : J∞E → M is the

inverse limit of the tower of jet bundles M ← E · · · ← JkE ← Jk+1E ← · · ·under the standard projections πk

l : JkE → J lE, k > l. The limit J∞E is atopological space: a basis for a topology on J∞E is the collection of all sets of theform (π∞

k )−1(W ), in which W is an open subset of JkE. We describe the spaceJ∞E locally by sequences

(2.11) (xi, uα, uα

i1 , ..., uαi1i2...ik

, ...), 1 ≤ i1 ≤ i2 ≤ · · · ≤ ik ≤ n,

obtained from the standard coordinates on the finite–order jet bundles JkE. Wewould like to consider J∞E as a manifold. However, since (2.11) implies that


the dimension of J∞E must be infinite, we need to re-define all of the standarddifferential geometry notions in this new context, see [4, 30, 32]:

Let π∞k : J∞E → JkE denote the canonical projection from J∞E onto JkE.

We say that a function f : J∞E → R is smooth if it factors through a finite–orderjet bundle, that is, if f = fk ◦ π∞

k for some smooth function fk : JkE → R. Real-valued smooth functions on J∞E are also called differential functions, see [36].More generally, if P is some finite-dimensional manifold, a function f : J∞E → Pis smooth if there exist k ≥ 0 and a smooth function fk : JkE → P such thatf = fk ◦ π∞

k , and a function f : P → J∞E is smooth if for any finite-dimensional

manifold Q and any smooth map g : J∞E → Q the composition g ◦ f is smooth.Finally, if F → N is another fiber bundle, a function Φ : J∞E → J∞F is smooth iffor every smooth map g : J∞F → Q, the composition g ◦Φ : J∞E → Q is smooth.

Vector fields X on J∞E are derivations on the ring of smooth real-valuedfunctions on J∞E. In local coordinates, vector fields are formal series of the form

(2.12) X = Ai∂

∂xi+

∑k≥0

1≤i1≤···≤ik≤n

Bαi1...ik

∂

∂uαi1...ik

,

in which Ai, Bαi1...ik

are smooth functions on J∞E.Now, in analogy with the finite-dimensional case, any local section s : (xi) →

(xi, sα(xi)) of E lifts to a unique local section j∞(s) of J∞E called the infinite

prolongation of s. In coordinates, j∞(s) is the section

j∞(s)(xi) =

(xi, s

α(xi),∂sα

∂xi1

(xi) , ...,∂ksα

∂xi1 . . . ∂xik

(xi) , ...

).

Also, if V is a vector field on the base space M , the infinite prolongation, pr∞Vcan be defined as in (2.5): the action of pr∞V on a smooth function f on J∞E ata point σ = j∞(s)(p) ∈ J∞E, p ∈M , is given by

(2.13) pr∞V (σ) · f = V (p) · (f ◦ j∞(s)) .

Particularly important are the infinite prolongations of the partial derivatives ∂/∂xj .We obtain

(2.14) Dj = pr∞(

∂

∂xj

)=

∂

∂xj+ uα

j

∂

∂uα+ ui1j

∂

∂uαi1

+ ui1i2j∂

∂uαi1i2

+ · · · .

As before, the operators Dj are called total derivative operators. The infinite pro-longation pr∞V on J∞E of a vector field V on E is defined exactly as in (2.6),(2.7). The same formulae hold for pr∞V , the only difference being that in this casewe need to sum over all multi-indices J with #J ≥ 0 in (2.6).

Now we define the Cartan connection on J∞E: the horizontal lift of a vectorfield V on M —also called the total derivative in the V direction— is simply pr∞V .Locally, horizontal vector fields are linear combinations of the total derivatives Dj

defined in (2.14).It is possible to check, see [30, 32], that in contrast with finite-dimensional

jet bundles, pr∞[V1, V2] = [pr∞V1, pr∞V2] for all vector fields V1 and V2 on M ,

and therefore the Cartan connection determines, at least formally, an integrabledistribution on J∞E which we denote C. Locally, the maximal integral manifoldsof C are the graphs of the infinite prolongations j∞(s) of local sections s : M → E,see [32].


Finally, to complete this brief description of differential geometry on J∞E,we consider differential forms: by definition, smooth differential forms on J∞Eare the pull–backs of differential forms on JkE by the projections π∞

k . Thus, anydifferential k–form ω on J∞E may be written in canonical coordinates as a finitelinear combination of terms

(2.15) Adxi1 ∧ · · · ∧ dxip ∧ duα1j1...jp1

∧ · · · ∧ d uαq

l1...lpq,

in which p+ q = k and A is a smooth function on J∞E.

2.3. Differential equations. Consider a k th order differential equation (ormore generally, a k th order system of DEs)

(2.16) Ξ

(xi, u

α,∂uα

∂xi, ...,

∂kuα

∂xi1 . . . ∂xik

)= 0 .

This equation defines a locus in the jet bundle JkE. We restrict this locus toa submanifold S(k) of JkE, and we assume that the function Ξ is smooth on aneighbourhood of S(k). We further ask S(k) to be a sub–bundle of JkE, so thatit fibers over the space M of independent variables. This sub-bundle is called theequation manifold of Ξ = 0.

The full equation manifold S(∞) of Ξ = 0 is a sub-bundle of the infinite jetspace J∞E constructed thus: The prolongations S(k+1), S(k+2), . . . of S(k) aredefined by total differentiation of (2.16), so that, for instance,

S(k+1) = {jk+1(s)(xi) ∈ Jk+1E : jk(s)(xi) ∈ S(k)

and (DiΞ)(jk+1(s)(xi)) = 0 for all i = 1, 2, . . . , n} .

We assume that the tower M ← S(k) ← S(k+1) ← · · · is well defined, that is, eachS(l+1) (l ≥ k) is a submanifold of J l+1E which fibers over S(l). We then defineS(∞) as the inverse limit of the tower M ← S(k) ← S(k+1) ← · · · . Locally, S(∞)

is the set of infinite jets in J∞E satisfying (2.16) and all its (total) differentialconsequences.

A local solution of Equation (2.16) is a local section of S(k) which is the kthprolongation of a local section s : (xi) → (xi, u

α(xi)) of E. A local smooth solutionof Equation (2.16) is a local section of of S(∞) which is the infinite prolongation ofa local section s : (xi) → (xi, u

α(xi)) of E.If σ = j∞(s)(p) is a point in S(∞), we define the horizontal subspace Cσ(S(∞))

of TσS(∞) as Cσ(S(∞)) = TσS

(∞)∩Cσ, in which Cσ is the horizontal subspace of theCartan distribution on J∞E at σ. Thus, there exists a connection on S(∞) —alsocalled the Cartan connection— obtained by restricting the Cartan connection onJ∞E. Locally, the horizontal vector fields on S(∞) are linear combinations of therestriction to this manifold of the total derivatives Di defined in (2.14).

Finally, we make the important remark that vector fields and differential formson S(k) and S(∞) are defined via pull-back by the canonical inclusions ι(k) : S(k) ↪→JkE and ι : S(∞) ↪→ J∞E. For example, the contact ideal of S(k), I(S(k)), is thecontact ideal Ik(E) of JkE pulled-back to S(k) by the inclusion ι . Examples ofcontact ideals of equations of interest for mathematical physics appear in [23].

Example 2.6. Let ut = F (x, t, u, ..., uxk) be an evolution equation in two

independent variables. Natural coordinates on S∞ are (x, t, u, ux, ..., uxx...x, ...), as


derivatives of u with respect to t may be replaced by expressions containing only x–derivatives by using the differential consequences of ut = F . The horizontal vectorfields on S(∞) are linear combinations of the total derivatives (2.4) restricted toS(∞), that is,

(2.17) Dx =∂

∂x+

∞∑i=0

ui+1∂

∂uiand Dt =

∂

∂t+

∞∑i=0

Dix(F )

∂

∂ui,

and the basic contact forms (2.2) restricted to S(∞) become

θαJ = d uαJ − uJ xdx−DJ (F )dt ,

in which J = (j1, . . . , jk) is a multi-index, uJ x abbreviates uxj1xj2

...xjkx where each

xji is either x or t, and DJ = Dxj1Dxj2

. . .Dxjk. These formulae are at the basis

of the algebraic approach to integrability of evolutionary PDEs, see for instance[33, 34].

3. A glimpse of Gromov’s h-principle

We generalize the geometric definition of a differential equation: a differentialrelation R of order k is an arbitrary sub-manifold of JkE. In particular, we donot insist that R fibers over the base space M . For example, consider M = Rn,E = Rn ×Rq, and take

R=

⎧⎪⎨⎪⎩(x1, . . . , xn, u1, . . . , uq, u1

xi, . . . , uq

xi)i=1,...n : rank

⎛⎜⎝ u1x1

. . . u1xn

......

...uqx1

. . . uqxn

⎞⎟⎠=n

⎫⎪⎬⎪⎭ ,

so that R is an open subset of J1E. Let us identify a function f : M → Rq,f(xi) = (uα(xi)) with the section sf : (xi) → (xi, u

α(xi)) of E. We see that thefunction f is an immersion if and only if the section sf solves the differential relationR, that is —again generalizing from the previous section— if and only if the firstorder prolongation of sf , j

1(sf ), is contained in R.Now, we can also consider formal solutions to a given differential relation R ⊆

JkE. We simply state that a formal solution to R is a section s : M → JkE suchthat s(M) ⊂ R. We now define, after [20]:

Definition 3.1. A differential relationR ⊆ JkE satisfies Gromov’s h-principleif every formal solution s of R is homotopic (within R) to a genuine solution to R,that is, if there exists a one-parameter family of sections st : M → JkE, 0 ≤ t ≤ 1,such that st(M) ⊂ R for all t, s0 = s, and s1 : M → JkE is of the form s1 = jk(s′)for a section s′ of E.

A section of JkE such as s1 in Definition 3.1 is called an holonomic section ofJkE. Note that in the special case of a kth order differential (system of) equation(s)

(3.1) Ξ

(xi, u

α,∂uα

∂xi, ...,

∂kuα

∂xi1 . . . ∂xik

)= 0,

a formal solution would be simply a section s : M → S(k), and we would say that(3.1) satisfies Gromov’s h-principle if there exists a one-parameter family of sectionsst : M → Sk, 0 ≤ t ≤ 1, such that s0 = s and s1 is a local solution of Equation(3.1).


A very readable exposition of Gromov’s h-principle appears in Y. Eliashbergand N. Mishachev’s book [20]. As stated in this reference, it is not usually thecase that non-trivial PDEs satisfy the h-principle, but on the other hand, there aresome geometrically significative differential relations which do obey it. Eliashbergand Mishachev proved in 2001 (see [20] and references therein) an approximationtheorem for sections of jet bundles from which several “h-principle results” follow.We state it below, after making a few notational remarks:

Let us fix a metric dist : JkE × JkE → R on JkE. (For instance, we couldprescribe a Riemannian metric tensor on JkE and use it to define a metric function).We define an isotopy ψt : M → M , 0 ≤ t ≤ 1, as a one-parameter family ofdiffeomorphisms of M such that ψ0 = idM , and we say that the isotopy ψt isδ-small if dist(ψt(p), p) < δ for all p ∈M and 0 ≤ t ≤ 1. We also recall that a sub-polyhedron P ⊂ M is a subcomplex of a triangulation of M , and that P is properif no simplex of P is of dimension equal to the dimension of M . (Triangulations ofnot necessarily compact manifolds are discussed in Whitney’s treatise [51]).

Theorem 3.2 (Holonomic approximation theorem, [20]). Let P be a propersub-polyhedron of M and let s : U → JkE be a local section, in which U is anopen neighborhood of P . For arbitrary δ, ε > 0 there exists a δ-small isotopy ψt :

M →M , 0 ≤ t ≤ 1, and a holonomic section s : U → JkE, in which U is an openneighborhood of the image ψ1(P ), such that ψ1(P ) is contained in the domain ofthe original section s and

dist (s(p) , s(p)) < ε for all p ∈ U .

It is usually assumed that M is compact, but this restriction is not necessary,see [20]. In order to apply this result to differential equations, let us agree that foran ε-approximate solution to S(k) we mean a holonomic section σ of JkE such thatσ solves the open differential relation Rε, in which Rε is an open neighborhood ofS(k) satisfying dist(p, S(k)) < ε for all p ∈ Rε. We have [20]:

Corollary 3.1. Let S(k) ⊂ JkE be a closed hypersurface, P ⊂ M a propersub-polyhedron of M and assume that s : U → S(k) ⊂ JkE is a formal solution toS(k), in which U is an open neighborhood of P . For arbitrary δ, ε > 0 there exista δ-small isotopy ψt : M → M , 0 ≤ t ≤ 1, and an ε-approximate solution jk(s) toS(k), such that ψ1(P ) is contained in the domain of s and

dist(jk(s)(p) , s(p)

)< ε for all p ∈ U ,

in which U is an open neighborhood of ψ1(P ) contained in the domain of jk(s).

In other words, given a formal solution to S(k) with domain D ⊆M , it is alwayspossible to deform D slightly to a domain D′ ⊆ M , and to find an ε-approximatesolution to S(k) with domain D′:

Example 3.3. Let F (x1, . . . , xn, u, ux1, . . . , uxn

) = 0 be a scalar first orderequation with (x1, . . . , xn) in an open set U ⊂ Rn, and suppose we wish to solvethe Cauchy problem with initial data u|Γ = g, in which Γ is a subset of ∂U . Thisproblem is treated in classical books such as [21]. First of all, we use the datau|Γ = g to construct admissible initial data on a subset of J1(Rn ×R): we fix apoint x0 ∈ Γ, and we determine the values of the derivatives uxi

for y ∈ Γ near x0

[21]. We obtain an (n − 1)-dimensional submanifold Γ′ contained in the equationmanifold S(1) of F = 0. We can now take an open neighborhood of Γ′ in S(1) and


any section s extending a section s determined by Γ′. That such a section s existsis guaranteed by the smooth Tietze extension theorem [1, p. 380]. This section isa formal solution to the equation F = 0, which extends the initial data Γ′. Thelast corollary implies that for any ε > 0 there exists an ε-approximate solution toF = 0 arbitrarily near a “small perturbation” of the original set Γ.

4. Symmetries of differential equations

Let us assume for a moment that Ξ(xi, u, . . . ) = 0 is a scalar partial differentialequation. Intuitively, a symmetry of Ξ = 0 is any function G such that for anysolution u(xi) to Ξ = 0, the deformed function u(xi)+ τG(u(xi)) is also a solution,to first order in τ . Symmetries may be local (i.e., G may be a smooth function onsome finite or infinite-dimensional jet bundle) or nonlocal (i.e., G may depend forinstance on integrals of u). Local symmetries of differential equations are classical[5, 10, 36, 37], and therefore we introduce them very quickly, just to fix ournotation. We then concentrate in nonlocal symmetries following [45, 44, 25, 26].

4.1. Local symmetries.4.1.1. Classical symmetries. Let Ξ(xi, u

α, . . . , uαJ ) = 0 be an nth order (system

of) partial differential equation(s). A vector field V on the space E of independentand dependent variables,

V =n∑

i=1

ξi(xj , uβ)

∂

∂xi+

m∑α=1

ϕα(xj , uβ)

∂

∂uα,

is a classical symmetry of Ξ = 0 if the equations

(4.1) pr(n)V (Ξ) = 0

hold whenever uα(xi) is a solution to Ξ = 0, see [5, 10, 32, 36].The flow of the vector field V on the space E is determined by the system of

equations

(4.2)d xi

d τ= ξi(xj , u

β),d uα

d τ= ϕα(xj , u

β),

and it can be rigorously proven [36] that if Equation (4.1) holds and the systemΞ = 0 satisfies some mild technical conditions, then for each value of the parameterτ the transformation xi → xi(τ ), u

α → uα(τ ) sends solutions of Ξ = 0 to solutionsof the same system.

Example 4.1. In [36, pp. 117–120], P.J. Olver computes all classical symme-tries of the heat equation ut = uxx. One of them is

V = 4t2∂

∂t+ 4tx

∂

∂x− (x2 + 2t)u

∂

∂u.

The flow of this vector field is calculated via Equations (4.2). We obtain that ifu(x, t) solves the heat equation, so does the function

(4.3) u′(x, t) =1√

1 + 4τtexp

(−τx2/(4τt+ 1)

)u

(x

1 + 4τt,

t

1 + 4τt

).

In particular, if the “seed” solution u is a non-zero constant, u(x, t) = c, we concludethat the function

(4.4) u′(x, t) =c√

1 + 4τtexp

(−τx2

4τt+ 1

)


also solves the heat equation. As observed in [36], this is, up to translations andconstants, the fundamental solution to the heat equation. This interesting observa-tion may seem slightly counterintuitive at first: it looks like we are “creating” heat(the solution u′(x, t)) from equilibrium (the solution u(x, t) = c) as we move alongτ ! But of course this is not so, because τ is not “physical time”.

P. Olver’s observation has been the starting point of several investigations.Notably, M. Craddock and A. Dooley [13] and then M. Craddock and E. Platen[14] have characterized all functions f(x) for which the equations ut = uxx+f(x)ux

and ut = xuxx + f(x)ux admit point symmetries carrying constant solutions tofundamental solutions.

4.1.2. Generalized symmetries. We consider first order differential operators onJ∞E of the special form

(4.5) V =

n∑i=1

ξi∂

∂xi+

m∑α=1

ϕα ∂

∂uα,

in which ξi and ϕα are smooth functions on J∞E. Geometrically, a differentialoperator such as (4.5) is a vector field along the projection π∞

E , that is, a smoothmap V : J∞E → TM such that for all σ ∈ J∞E, V (σ) ∈ Tπ∞

E (σ)E.

Following established terminology, [5, 36, 37], we call V a generalized vectorfield. V is a generalized symmetry of the system Ξ = 0 if and only if pr∞V (Ξ) = 0whenever uα(xi) is a solution of Ξ = 0, in which pr∞V is the infinite prolongation ofV defined in Section 2.2. We can split pr∞V using the Cartan connection of J∞Ein horizontal and vertical parts, pr∞V = VH + V ert(V ). Since VH is a finite linearcombination of total derivatives, it always satisfies VH(Ξ) = 0 on (local smooth)solutions, and therefore it is enough to consider generalized symmetries V such thatpr∞V is a vertical vector field. Such a V must be of the form [5, 30, 32, 36]

(4.6) V =

m∑α=1

Gα ∂

∂uα,

for some m-tuple of differential functions G = (Gα), so that its infinite prolongationis simply

pr∞V =

m∑α=1

∑#J≥0

DJGα ∂

∂uαJ

,

where DJ = Dj1Dj2 . . . Djk for J = (j1, j2, . . . jk). The symmetry conditionpr∞V (Ξ) = 0 for generalized vector fields such as (4.6) says precisely that the “in-finitesimal deformation” uα(xi) → uα(xi) + τGα(uα(xi)) is —to first order in τ—a solution of Ξ = 0 whenever uα(xi) is a solution of Ξ = 0.

At least at a formal level, a generalized symmetry VG of a system of equationsΞ = 0 transforms solutions into solutions: if uα

0 (xi) is a solution to Ξ = 0, weconsider the Cauchy problem for the “integral curve” of the generalized vector fieldV =

∑mα=1 G

α ∂/∂uα through uα0 (xi), namely,

(4.7)∂uα

∂τ= Gα ; uα(xi, 0) = uα

0 (xi) .

Then as proven in [36], for any value of the parameter τ , a solution uα(xi, τ ) tothe Cauchy problem (4.7) —assuming it exists, see [5, 10, 36]— is also a solutionto the original system Ξ = 0.


4.1.3. Internal symmetries. An infinitesimal contact transformation is a vectorfield X on JkE such that X(Ik(E)) ⊆ Ik(E), that is, the Lie derivative of anycontact form with respect to X is again a contact form. An infinitesimal contacttransformation X is an external symmetry of a system of differential equationsΞ = 0 if X(Ξ) = 0 on solutions to Ξ = 0. An internal symmetry of a systemof equations Ξ = 0 of order k is a vector field W on S(k) (so that the conditionW (Ξ) = 0 holds) such that W (I(S(k))) ⊆ I(S(k)). These symmetries were firstconsidered by E. Cartan, see [5, 23, 32, 37] and references therein.

Certainly, if W is an external symmetry, then the restriction of W to S(k) isan internal symmetry. On the other hand, the following basic result holds [5]:

Theorem 4.2. Let Ξ = 0 be a system of equations of order k and let W be avector field on S(k). If W is an internal symmetry and V = π(W ) is the projectionof W to the bundle E, then V is a generalized symmetry of the system Ξ = 0, andmoreover, W = (pr(k)V )|S(k) .

The converse to Theorem 4.2 is not necessarily true. In order for a generalizedsymmetry V to determine an internal symmetry, we must check the contact con-dition pr(k)V (I(S(k))) ⊆ I(S(k)), which is not trivial since pr(k)V may depend onderivatives of order (k + 1) or higher. A subtle result proven in [5] states that if asystem of partial differential equations Ξ = 0 satisfies a technical condition (dubbedthe “descent property”) then it is indeed true that every internal symmetry of Ξ = 0arises from a generalized symmetry.

Now, one of the important observations of [5] is that the descent property isactually a common property, and so it would seem that genuine internal symmetriesare quite hard to come by. However, as it will be briefly discussed in Section4.3, nonlocal symmetries of partial differential equations are a natural source of“generalized” internal symmetries.

4.2. Nonlocal symmetries.4.2.1. Basic definitions. Intuitively, a function G depending on x, t, u, a finite

number of x–derivatives of u, and for example integrals of u, is a “nonlocal sym-metry” of a scalar equation Ξ(x, t, u, . . . ) = 0 if for any solution u(x, t) of Ξ = 0,the function u(x, t) + τG(u(x, t)) is also a solution to first order in τ , precisely asin the generalized symmetry case.

These symmetries are indeed important for differential equations. For instance,G. Bluman and his collaborators have used them to find nonlocal transforma-tions relating nonlinear PDEs (a recent summary of their work appears in [6])and Sokolov and Svinolupov [48] have used them to classify integrable second or-der evolutionary PDEs. They have been also used to derive Backlund/Darbouxtransformations [46, 45, 25], and to obtain highly nontrivial explicit solutions tononlinear PDEs [22, 45, 44, 46].

A completely general (i.e., without a priori restrictions on the form nonlocalobjects may take, as it happens for instance in the recent textbook [10] on appliedaspects of symmetry theory) geometric formulation of nonlocal symmetries hasbeen developed by J. Krasil’shchik and A. Vinogradov, see [49, 30, 31, 32] andreferences therein. In this section we present a version of their theory following [45,44, 25, 26]. We remark that other approaches to nonlocal symmetries (of formal,algebraic or heuristic nature) are possible, see for instance [2, 3, 6, 10, 38, 39],


but we believe the Krasil’shchik-Vinogradov theory is the most satisfying approachto nonlocalities available today (some comments along these lines are also in [39]).

We begin with a classical example appearing already in [49]:

Example 4.3. We consider Burgers’ equation ut = uxx + uux. The expression

(4.8) G = (2Sx − uS) e−1

2

∫u dx

,

in which S is any solution to the heat equation St = Sxx, satisfies the followingcondition: if u is a solution to Burgers’ equation, the “deformation” u+ τG is alsoa solution to first order in τ , that is, G formally satisfies the linearized Burgersequation. We make this observation rigorous by considering an extra dependentvariable γ such that γx = u and γt = ux + (1/2) u2. Then, we can write (4.8)as G = (2Sx − uS) exp(− 1

2γ), so that G becomes “local” and could be perhapsconsidered as the characteristic of a local symmetry for the “augmented” system

(4.9) ut = uxx + uux ; γx = u ; γt = ux + (1/2) u2 .

But then, in order to formalize this idea, we also need to consider the infinitesimalvariation of γ as u changes to u + τG ! This observation led Vinogradov andKrasil’shchik to develop their theory of coverings of PDEs.

Definition 4.4. Let N be a non-zero integer. An N -dimensional covering πof a system of partial differential equations Ξ = 0, is a pair

(4.10) π =({γb : b = 1, . . . , N} ; {Xib : b = 1, . . . , N ; i = 1, . . . , n}

)of variables γb and smooth functionsXib depending on xi, uα, γb and a finite numberof partial derivatives of uα, such that the equations

(4.11)∂γb

∂xi= Xib

are compatible whenever uα(xi) is a solution to Ξ = 0.

We usually write π = (γb ; Xib) instead of (4.10). Generalizing Example 4.3,we consider the variables γb as new dependent variables, the “nonlocal variables”of the theory, and we interpret Equations (4.11) as the equations stating how thevariables γb relate to the original variables uα, as in (4.9).

Our definition of nonlocal symmetries is the following (Compare [32, pp. 249-250]):

Definition 4.5. Let Ξ = 0 be a system of partial differential equations, andlet π = (γb; Xib) be a covering of Ξ = 0. A nonlocal π–symmetry of Ξ = 0 is ageneralized symmetry

X =∑i

ξi∂

∂xi+∑α

φα ∂

∂uα+∑b

ϕb ∂

∂γb

of the augmented system

(4.12) Ξ = 0 ;∂γb

∂xi= Xib .


Thus, in order to find nonlocal π-symmetries, we proceed exactly as in the localcase: we check the conditions

(4.13) pr∞X (Ξ) = 0 and pr∞X

(∂γb

∂xi−Xib

)= 0

whenever uα(xi) and γb(xi) are solutions to the augmented system (4.12). As werecalled in 4.1.2, in order to capture all generalized symmetries of a given systemof equations it is enough to consider evolutionary vector fields X of the form

(4.14) X =

m∑α=1

Gα ∂

∂uα+

N∑b=1

Hb ∂

∂γb.

Note that the first equation in (4.13) depends only on the system Ξ = 0 andon the “nonlocal vector field” Gα ∂/∂uα. J. Krasil’shchik and A. Vinogradov [30,31, 32, 49] call G = (Gα) the π–shadow of the nonlocal π–symmetry (4.14).Interestingly, most (if not all) work on nonlocal symmetries carried out outside theKrasil’shchik–Vinogradov framework is only about shadows.

Proposition 4.1. If uα0 (x

i) and γb0(x

i) are solutions to the augmented system(4.12), the solution to the Cauchy problem

∂uα

∂τ= Gα,

∂γb

∂τ= Hb ;

uα(xi, 0) = uα0 (x

i), γb(xi, 0) = γb

0(xi) ;

is a one–parameter family of solutions to the augmented system (4.12). In partic-ular, nonlocal π–symmetries send solutions to the system Ξ = 0 to solutions of thesame system.

Several nontrivial examples of nonlocal symmetries appear in [49, 22, 24, 32,42, 43, 45, 44, 46, 25, 26] and works cited therein. Computations are generallyquite demanding, but at least we are allowed to replace all derivatives of γb ap-pearing in (4.13) by means of Equations (4.11) and their differential consequences(see [36, p. 292]), so that we can assume without loss of generality that the coef-ficients Gα and Hb of the vector field (4.14) depend only on xi, uα, finite numbersof derivatives of uα, and the new variables γb.

4.2.2. Reconstructing shadows. We begin with an example:

Example 4.6. We complete Example 4.3 on Burgers’ equation ut = uxx+uux.Recall that we have considered an extra dependent variable γ such that γx = u =X11 and γt = ux + (1/2) u2 = X21. These two equations are of course compatibleon solutions to Burgers’equation, and so we have a one-dimensional covering. Nowset G = (2Sx − uS) exp(−(1/2)γ), and determine the infinitesimal variation H1 of

γ as u changes to u + τG : γτx = γxτ = uτ = (2Sx − uS) e−12γ = (2S e−

12γ)x, so

that γτ = 2S exp(− 12γ), and therefore we can define H = 2S exp(− 1

2γ). We cancheck that the vector field

X = G∂

∂u+H

∂

∂γ

is a generalized symmetry of the equations

ut = uxx + uux, γx = u, γt = ux + (1/2) u2,


and therefore X is a nonlocal symmetry for Burgers’ equation. Following [49], wenote that solutions u(x, t) which are invariant under this nonlocal symmetry satisfyG(u(x, t)) = 0, and so the equation

(4.15) 2Sx − uS = 0

must hold. This is the Cole–Hopf transformation between the nonlinear Burgersequation ut = uxx + uux and the linear heat equation St = Sxx !

In some cases, see for instance [45], (4.14) is a classical symmetry of the aug-mented system (4.12), but there exist examples for which the vector field (4.14) is agenuine generalized symmetry, see [22, 42, 43, 44, 25, 26] and Section 4.3 below.

Shadows can be always completed into genuine nonlocal symmetries as we didin Example 4.6, if we extend the foregoing theory to infinite-dimensional coveringsas in [30, 31, 32]:

Proposition 4.2. Let π = (γb; Xib) be a covering of the system Ξ = 0. Forany π-shadow G there exists a further covering π′ and a π′-nonlocal symmetry whichextends the π-shadow G.

Proof. This proof is modelled after [30]. We know that uατ = Gα and that

γbxi = Xib. As in Examples 4.3 and 4.6, we need to find the infinitesimal variations

of the nonlocal variables γb. We simply set

∂γb∂τ

= γb1,

in which γb1 are new symbols for nonlocal variables. Since once we add these new

symbols we need to find infinitesimal variations for them, we consider an infiniteset of variables γb

l , l = 0, 1, 2, . . . , and we set

∂γbl

∂τ= γb

l+1,

with the convention that γb0 = γb. Now it remains to find the relations among the

nonlocal variables γbl and the local dependent variables uα. We compute formally

“on solutions”:∂γb

l

∂xi=

∂γbl−1,τ

∂xi= · · · = ∂γb

τ...τ

∂xi=

∂lXib

∂τ l,

and this last derivative can be found by applying l times the τ -derivative operator

Dτ =∑

DJGα ∂

∂uαJ

+∑

γbl+1

∂

∂γbl

to Xib. We conclude that the infinite compatible system Ξ = 0, γbl,xi = Dl

τ (Xib)

possesses the “generalized symmetry”

X =

m∑α=1

Gα ∂

∂uα+∑b,l

γbl+1

∂

∂γbl

.

The vector field X is a bona fide nonlocal π′-symmetry of Ξ = 0 defined on theinfinite-dimensional covering

π′ =({γb

l : b = 1, . . . , N ; l ≥ 0}; {Dlτ (Xib) : i = 1, . . . , n; b = 1, . . . , N ; l ≥ 0}

)of Ξ = 0 which extends the shadow G. �


Other reconstruction results have been obtained by N. Khor’kova and K. Kiso,see [30, 32] and references therein. Of course this last proposition, while the-oretically satisfactory, is only of relative practical value. It is quite interestingthat in explicit examples shadows can be reconstructed into nonlocal symmetrieson finite-dimensional coverings, so that the Krasil’shchik–Vinogradov theory doeshave immediate applicability.

4.2.3. Nonlocal symmetries and exterior differential systems. The main obser-vation to be explored here is that the system of compatible equations (4.11) definesa completely integrable Pfaffian system I = {Γb} by means of

(4.16) Γb = dγb −n∑

i=1

Xib dxi .

More precisely, compatibility of the system

(4.17)∂γb

∂xi= Xib

is equivalent to the complete integrability condition

dΓb ≡ 0 mod {Γ1, . . . ,ΓN}

whenever uα(xi) is a solution to the system Ξ = 0. Motivated by this remark, wegeneralize our nonlocal setting as follows:

Definition 4.7. An N -dimensional I-covering of a system of partial differen-tial equations Ξ = 0 is an exterior differential system I over N = J∞E ×RN suchthat I is completely integrable whenever uα(xi) is a solution to the system Ξ = 0.

We can also generalize our notion of a nonlocal symmetry:

Definition 4.8. Let I be an N -dimensional I-covering of a system of partialdifferential equations Ξ = 0 defined on the manifold N = J∞E × RN . An I-nonlocal symmetry of Ξ = 0 is a generalized vector field X over N such that

(4.18) pr∞X(Ξ) ≡ 0 mod I

and

(4.19) Lpr∞XI ≡ 0 mod I,

whenever uα(xi) is a solution to the system Ξ = 0, in which Lpr∞X indicates Liederivative with respect to pr∞X.

Example 4.9. The Degasperis-Procesi (DP) equation [16] is

(4.20) m+ uxx − u = 0 mt +mx u+ 3mux = 0 .

This equation, together with the Camassa-Holm equation [9] to be reviewed in thefollowing section, exhaust the list of integrable equations in the family

ut − uxxt + (b+ 1) uux = b uxuxx + uuxxx, b ∈ R,

see [16, 17]. We can extract an I-covering of Equation (4.20) from the Degasperis-Holm-Hone integrability analysis appearing in [17]. Indeed, in that paper is shown


that the system of equations

− γ + γxx + λm+ γ3 + 3 γ γx = 0(4.21)

−γt − γ ux − γx u−2 γxγ

λ+ uxx −

γxxλ

= 0(4.22)

−δx + γ = 0(4.23)

−δt + ux − u γ − γxλ− γ2

λ= 0(4.24)

is compatible whenever m and u solve (4.20). The function γ can be consideredas a “pseudo-potential” for the Degasperis-Procesi equation, but it is importantto note that it falls outside the usual definition of a Wahlquist-Eastbrook pseudo-potential (see [50] and the definitions of pseudo-potentials appearing for instancein [24, 25, 26]). It follows that (4.21)–(4.24) does not determine a covering inthe sense of Vinogradov and Krasil’shchik, see Definition 4.4 and [32, Chapter 5].It does determine an I-covering, since it is a standard fact [8] that a compatiblesystem such as (4.21)–(4.24), can be written as a completely integrable exteriordifferential system. I-nonlocal symmetries of the DP equation are

(4.25)∂

∂x+ F2(λ)

∂

∂δ;

∂

∂t+ F1(λ)

∂

∂δ;

∂

∂δ,

in which F1, F2 are arbitrary functions of λ. It remains as an open problem to findfurther nonlocal symmetries of the DP equation depending on α and δ, since theones appearing in (4.25), while confirming that Definition 4.8 is not vacuous, areof limited practical utility.

Remark 4.10. There exists a deep theory of symmetries of integrable equations(the adjective “integrable” is understood here as meaning that the equation at handis the integrability condition of a one-parameter family of overdetermined linearproblems) based on loop groups, see R. Palais’ well-known review article [40]. Itwould appear that the precise relationship between the symmetries discussed byPalais and nonlocal symmetries has not been considered in detail as yet. We domention, however, the interesting paper [2], in which V.E. Adler considers (shadowsof) nonlocal symmetries for a general class of integrable equations.

4.3. The Camassa-Holm and associated Camassa-Holm equations. Inthis section we consider nonlocal symmetries of the Camassa-Holm (CH) equation[9]

2ux uxx + uuxxx = ut − uxxt + 3ux u,

which we write as a system of equations for two dependent variables m and u,

(4.26) m = uxx − u, mt = −mx u− 2mux,

and of the associated Camassa-Holm (ACH) equation [47]

(4.27) pT = −p2uy, u =−p22−(pTp

)y

p .

We will explain the precise relation between (4.26) and (4.27) momentarily. Werecall from Section 1 that nonlocal symmetries will be used in Section 5: they willallow us to find Darboux transformations and explicit solutions. Other applicationsappear in [24, 26, 45].

Results reported herein are mainly joint work with R. Hernandez Heredero(Universidad Politecnica de Madrid).


Proposition 4.3. The following systems of equations are compatible on solu-tions to the Camassa-Holm equation (4.26):

γx = m− 1

2λγ2 +

1

2λ, γt = λ

(ux − γ − 1

λuγ

)x

;(4.28)

δx = γ, δt = λ

(ux − γ − 1

λuγ

);(4.29)

βx = m eδ/λ, βt = eδ/λ[−(1/2) γ2 + (1/2)λ2 − um

].(4.30)

This proposition is proven in [42] using a geometric approach explained brieflyin Remark 4.11 below, but of course it can be checked directly. Its importance forus rests on the fact that the compatible system of equations (4.28)–(4.30) yields acovering of the Camassa-Holm equation.

Remark 4.11. Equations (4.28) for the pseudo-potential γ yield the standardscalar Lax pair for the Camassa-Holm equation, [9], if we set γ = 2λ ln(ψ)x, see forinstance [24]. From a geometric point of view, quadratic pseudo-potentials such as(4.28) arise thus [42, 24]: the Camassa-Holm equation describes pseudo-sphericalsurfaces, that is, generic solutions to CH determine Riemannian metrics of Gaussiancurvature K = −1 on (open subsets of) their domains. Let u(x, t) be a solutionto CH, and assume that the pseudo-spherical metric g(u(x, t)) is determined bya moving coframe {ω1(u(x, t)), ω2(u(x, t))} and that the corresponding connection1-form is ω3(u(x, t)). Then, as explained by S.S. Chern and K. Tenenblat [12],there exists a new coframe {θ1, θ2} and a new connection form θ3 such that

dθ1 = 0, dθ2 = θ2 ∧ θ1, and θ3 + θ2 = 0 .

The forms θα are related to the forms ωα (we suppress the explicit dependence onsolutions for simplicity of notation) via

θ1 = ω1 cos ρ+ ω2 sin ρ, θ2 = −ω1 sin ρ+ ω2 cos ρ, θ3 = ω3 + dρ,

in which ρ is a solution to the Pfaffian system

(4.31) ω3 + dρ− ω1 sin ρ+ ω2 cos ρ = 0,

and, if {v1, v2} is the orthonormal moving frame dual to the moving coframe{θ1, θ2}, then the integral curves of the vector fields v1 and v2 are, respectively,geodesics and horocycles of M (see [12]). It follows, therefore, that the function ρsatisfying Equation (4.31) is precisely the angle connecting an arbitrary orthonor-mal framing with an orthonormal framing along geodesics. The key observationwe make now is that if we perform the change of variables γ = tan(ρ/2) (or,γ = cot(ρ/2)) the Pfaffian system (4.31) becomes a pair of Riccati equations. Thisis precisely how the quadratic pseudo-potential γ given by (4.28) arises from geom-etry.

Remark 4.12. The above remark allows us to conclude that quadratic pseudo-potentials are more fundamental from a geometric point of view than scalar Laxpairs. However, it should be also noted that no geometric interpretation similarto the one above appears to have been found for cubic pseudo-potentials such as(4.21) and (4.22).

We can classify all first order nonlocal π-symmetries of the CH equation. Notethat Equations (4.28)–(4.30) depend explicitly on the parameter λ. In order to


perform a complete classification, we find it necessary to assume that λ is alsoaffected by symmetry transformations. In other words, we extend the augmentedsystem (4.26), (4.28)–(4.30) with the equations

(4.32) λx = 0, λt = 0,

and we consider the four-dimensional covering π of CH determined by γ, δ, β andλ.

Theorem 4.13. The first order generalized symmetries of the augmented CHsystem (4.26), (4.28)–(4.32), represented by vector fields

(4.33) V = G1 ∂

∂m+G2 ∂

∂u+H1 ∂

∂γ+H2 ∂

∂δ+H3 ∂

∂β+K

∂

∂λ

in which Gα, Hb, and K are functions of x, t, m, u, their first derivatives, γ, δ,β, and λ are

V1 = (2mux + umx)∂

∂m− ut

∂

∂u+

(λ2

2− λu

2+ um− γ2

2− uγ2

2λ+ γux

)∂

∂γ

+ (λγ + uγ − λux)∂

∂δ− 1

2eδ/λ

(λ2 − 2um− γ2

) ∂

∂β,

(4.34)

V2 = mx∂

∂m+ ux

∂

∂u+

(λ

2+m− γ2

2λ

)∂

∂γ+ γ

∂

∂δ+ eδ/λm

∂

∂β,

(4.35)

V3 =∂

∂δ+

β

λ

∂

∂β,

(4.36)

V4 =∂

∂β,

(4.37)

V5 =eδ/λ(2mγ

λ+mx

)∂

∂m+ eδ/λγ

∂

∂u+ eδ/λm

∂

∂γ+ β

∂

∂δ+

(e2δ/λm+

β2

2λ

)∂

∂β,

(4.38)

V6 = A(λ)

(m

∂

∂m+ u

∂

∂u+ γ

∂

∂γ+ δ

∂

∂δ+ β

∂

∂β+ λ

∂

∂λ− t V1

),

(4.39)

where A(λ) is an arbitrary function. Consequently, these vector fields are nonlocalπ-symmetries of the CH equation.

This theorem appears in [25, 26]. The function A(λ) is included in V6 sinceit affects the way in which λ varies with the infinitesimal symmetry transforma-tion (4.39) and, in fact, it is of importance for the Lie algebra structure of nonlocalπ-symmetries [25, 26]. We also note that V1 and V2 are simply the generators ofshifts with respect to the independent variables: they are equivalent to ∂/∂t and−∂/∂x respectively.

Corollary 4.1. The commutation table of the six nonlocal symmetries (4.34)–(4.39) of the CH equation is that of Figure 1, whenever u,m, γ, δ, β, λ solve theaugmented CH system (4.26), (4.28)–(4.32).


V1 V2 V3 V4 V5 V6∗

V1 −B(λ)V1

V2

V3 − 1λV4

1λV5 B(λ)V3

V41λV4 V3 B(λ)V4

V5 − 1λV5 −V3

V6 A(λ)V1 − A(λ)V3 − A(λ)V4 −[A′(λ)B(λ)− A(λ)B′(λ)] λA(λ)

V6

Figure 1. Commutation table of the CH nonlocal symmetryalgebra. In V6

∗ we have used B(λ) instead of A(λ).

This corollary implies that the nonlocal π-symmetries (4.34)–(4.38) generatea five-dimensional Lie algebra G5 isomorphic to the direct sum of sl(2) and thecommutative Lie algebra generated by V1 and V2. In fact, if instead of V3, V4, V5

we usee = −

√2λV4, f =

√2λV5, h = −2λV3,

we find the commutators [h, e] = 2e, [h, f ] = −2f , [e, f ] = h. Now, if in additionto V1, . . . , V5 we consider also V6, we can check, following [26], that the infinite-dimensional Lie algebra generated by the nonlocal symmetries V1, . . . , V6 contains asemidirect sum of the loop algebra over sl(2,R) and the centerless Virasoro algebra:

Proposition 4.4. Let us define the vector fields

T 1n = −2λn+1V3, T 2

n = −√2λλnV4, T 3

n =√2λλnV5, and Wn = −λnL,

in which n ∈ Z and L = V6 with A(λ) = 1. Then, the following commutationrelations hold:

[T 1m, T 2

n ] = 2T 2m+n, [T 1

m, T 3n ] = −2T 3

m+n, [T 2m, T 3

n ] = T 1m+n ;

(4.40)

[Wm,Wn] = (m− n)Wm+n ;

(4.41)

[T 1m,Wn] = mT 1

m+n, [T 2m,Wn] = (m− 1

2)T 2

m+n, [T 3m,Wn] = (m+

1

2)T 3

m+n .

(4.42)

Remark 4.14. We stress the fact that the commutation relations of Corollary4.1 are valid only on shell . For example, the commutator of V5 and V6 is (we writeA, A′, A′′, instead of A(λ), A′(λ), A′′(λ) ):

(4.43) [V5, V6] = − eδ/λA′λx(m+ tmt)∂

∂m

− eδ/λ(A′′uλ2x + 2A′λxux + A′′tλ2

xut + 2A′tλxuxt + λxxA′u+ λxxA

′t ut)∂

∂γ,

so that [V5, V6] = 0 only after the augmented system of equations (4.26), (4.28)–(4.32) is imposed. Moreover, we note that [V5, V6] is not in the linear span ofV1, . . . , V6, unless we work on shell. This observation connects the foregoing theorywith internal symmetries: internal symmetries of a kth order equation form a Liealgebra on the equation manifold S(k) ⊂ JkE, but there is no reason why theyshould form a Lie algebra on the whole jet space JkE (see [5]). As (4.43) shows, the


same phenomenon appears in the realm of nonlocal symmetries. This is a reflectionof the fact that they are defined on coverings of the full equation manifold S(∞) ofthe differential equation being considered, and not on some “universal” jet space asit happens with local symmetries [39]. Thus, nonlocal symmetries really generalizethe class of internal symmetries.

Remark 4.15. In [29], S. Kouranbaeva proved that the group of diffeomor-phisms of the line (or the circle) appears as a configuration space for the Camassa-Holm equation (4.26). More precisely, Equation (4.26) is the geodesic spray of theweak H1 Riemannian metric on this group. Proposition 4.4 above explains whythis group of diffeomorphisms can be so naturally associated to CH.

Now let us consider the associated Camassa-Holm (ACH) equation (4.27), thatis,

pT = −p2uy, u =−p22−(pTp

)y

p .

This equation was first obtained by J. Schiff in [47] by applying the transformation

(4.44) p =√2m, d y = p dx− p u dt, and d T = d t

to the CH equation (4.26), and it has been studied for its own sake (i.e. withoutreference to its relation with the CH equation) in [27, 28].

Certainly, (4.44) is “nonlocal”: the new independent variable y is a “potentialvariable” which can be determined from x, t, p and u only through integration. Inparticular, (4.44) is not an invertible transformation, as explained in [47], and soCH and ACH are not equivalent equations. In fact, transformations such as (4.44)can change essentially the character of the nonlinearity of the initial equation, asremarked in [33, 34]. In particular, they can change the symmetry structure (see[36, pp.121, 122] for a discussion on this point in the context of Burgers’ equation).Motivated by these remarks, we compute nonlocal symmetries for ACH. Proposition4.3 yields:

Lemma 4.16.

(1) The associated Camassa-Holm equation (4.27) admits a pseudo-potentialγ determined by the compatible equations

(4.45) γy = − 1

2λpγ2 +

p

2+

λ

2p, γT =

γ2

2+

pTpγ + λu− 1

2λ2 .

(2) The associated Camassa-Holm equation (4.27) admits potentials δ and βdetermined by the compatible equations

(4.46) δy =γ

p, δT = λ

(−pT

p− γ

)and

(4.47) βy =p

2eδ/λ, βT =

1

2

(−γ2 + λ2

)eδ/λ .

And we also remark that transformation (4.44) implies that

(4.48) λy = λT = 0 .


Remark 4.17. The pseudo-potential equations (4.45) determine a Lax pair forthe associated Camassa-Holm equation. Indeed, setting γ = ψ2/ψ1 and replacinginto the first equation of (4.45) we find(

ψ1

ψ2

)y

=

(0 1

2λpp2 + λ

2p 0

)(ψ1

ψ2

),

precisely the y-part of the overdetermined linear problem for ACH obtained bySchiff in [47]. It appears that Schiff’s linear problem for ACH has not been studiedrigorously from the point of view of scattering/inverse scattering as yet.

Theorem 4.18. The first order generalized symmetries of the augmented ACHsystem (4.27), (4.45)–(4.48) are

W1 = −p2uy∂

∂p+ uT

∂

∂u

−(λ2

2− λu− γ2

2+ pγuy

)ux

∂

∂γ− λ (γ − puy)

∂

∂δ+

1

2eδ/λ(λ2 − γ2)

∂

∂β,

(4.49)

W2 = py∂

∂p+ uy

∂

∂u+

(λ

2p+

p

2− γ2

2λp

)∂

∂γ+

γ

p

∂

∂δ+

1

2eδ/λp

∂

∂β,

(4.50)

W3 = λ∂

∂δ+ β

∂

∂β,

(4.51)

W4 =∂

∂β,

(4.52)

W5 = 2eδ/λpγ∂

∂p

+ 2eδ/λλ (γ − puy)∂

∂u− eδ/λ

(λ2 − γ2

) ∂

∂γ− 2λ

(eδ/λγ − β

) ∂

∂δ+ β2 ∂

∂β,

(4.53)

W6 = A(λ)

(p∂

∂p+ 2u

∂

∂u+ 2γ

∂

∂γ+ 2(δ − λ)

∂

∂δ+ 2λ

∂

∂λ− 2T

∂

∂T+ y

∂

∂y

),

(4.54)

where A(λ) is an arbitrary function. Consequently, these vector fields are nonlocalsymmetries of the associated CH equation.

Corollary 4.2. The commutator table of the six nonlocal symmetries (4.49)–(4.54) of the associated CH equation is that of Figure 2 whenever u, p, γ, δ, β, λ solvethe augmented ACH system (4.27), (4.45)–(4.48).

The commutator table of the nonlocal symmetries W1, . . . ,W6 is not the sameas in the CH case, as it may have been expected from the comments on the non-equivalence of CH and ACH we made before. Interestingly, it is shown in [34] thattransformations such as (4.44) are invertible if we consider them as transformationson the space of variables u, ux, uxx, · · · . Invertibility is lost once we also consider theindependent variables x, t, as pointed out in [47]. It may be noted that the difference


W1 W2 W3 W4 W5 W6∗

W1 −2B(λ)W1

W2 B(λ)W2

W3 − W4 W5

W4 W4 2W3

W5 −W5 −2W3

W6 2A(λ)W1 −A(λ)W2 −2[A′(λ)B(λ)−A(λ)B′(λ)] λA(λ)

W6

Figure 2. Commutation table of the ACH nonlocal symmetryalgebra. In W6

∗ we have used B(λ) instead of A(λ).

between the commutation relations of Corollaries 3 and 4 appears precisely whenwe consider the scaling symmetries V6 and W6, and these symmetries do depend onthe independent variables x, t, y, T !

As in the CH case, the symmetries W1, ...,W5 generate a five-dimensional Liealgebra isomorphic to the direct sum of sl(2,R) and the two-dimensional commu-tative Lie algebra generated by W1, W2. Addition of W6 allows us to constructinfinite-dimensional Lie algebras of nonlocal symmetries to the ACH equation. In-stead of Proposition 4.4, in this case we obtain the standard semidirect sum of theloop algebra over sl(2,R) and the centerless Virasoro algebra:

Proposition 4.5. Let us define the vector fields

T 1n = −2λnW3, T 2

n = −λnW4, T 3n = λnW5, and Vn = − 1

2λnL,

in which n ∈ Z and L = W6 with A(λ) = 1. Then, the following commutationrelations hold:

[T 1m, T 2

n ] = 2T 2m+n, [T 1

m, T 3n ] = −2T 3

m+n, [T 2m, T 3

n ] = T 1m+n ;(4.55)

[Vm, Vn] = (m− n)Vm+n ;(4.56)

[T 1m, Vn] = mT 1

m+n, [T 2m, Vn] = mT 2

m+n [T 3m, Vn] = mT 2

m+n .(4.57)

5. Darboux transformations for the ACH equation

In this final section we construct Darboux transformations for the associatedCamassa-Holm equation. Darboux transformations for the standard Camassa-Holmequation are not considered in this paper because they are rather involved and wehave discussed them in detail elsewhere [25, 26].

First of all, we note that symmetries (4.49), (4.50) and (4.54) are extensions tothe covering of ACH determined by γ, δ and β of local symmetries, and that (4.51)and (4.52) are “vertical” symmetries, in the sense that they only transform someof the nonlocal variables. Thus, it is natural to focus our attention on (4.53).

We compute the flow of (4.53), so as to obtain a rule to generate solutions tothe associated Camassa-Holm equation

(5.1) pT = −p2uy, u =−p22−(pTp

)y

p .


The flow of (4.53) is determined by the system of equations

∂p

∂τ= 2eδ/λpγ,(5.2)

∂u

∂τ= 2λeδ/λ

(γ − p

∂u

∂y

),(5.3)

∂γ

∂τ= − eδ/λ

(λ2 − γ2

),(5.4)

∂δ

∂τ= − 2λ

(eδ/λγ − β

),(5.5)

∂β

∂τ= β2,(5.6)

with initial conditions u(y, T, 0) = u0, p(y, T, 0) = p0, γ(y, T, 0) = γ0, δ(y, T, 0) = δ0and β(y, T, 0) = β0. The solution to this initial value problem is determined by

β(τ ) =β0

1− τβ0,(5.7)

γ(τ ) =τ(λ2ω0 − ω0γ

20 + γ0β0

)− γ0

τβ0 − 1,(5.8)

δ(τ ) = δ0 − λ ln [(1− τβ0 + ω0γ0τ − τλω0) (1− τβ0 + ω0γ0τ + τλω0)] ,(5.9)

p(τ ) =p0 (1− τβ0 + ω0γ0τ − τλω0) (1− τβ0 + ω0γ0τ + τλω0)

(−1 + τβ0)2,(5.10)

in which ω0 = eδ0/λ. As shown in [25, 26], it is possible to obtain u(τ ) directlyfrom the ACH equation (5.1) without solving Equation (5.3). This equation isa consequence of the augmented ACH system (4.27), (4.45)–(4.48) and the flowequations (5.2) and (5.4)–(5.6):

Proposition 5.1. Assume that p, u and the functions γ, δ and β satisfy thecompatible equations (5.1), (4.45)–(4.48), (5.2) and (5.4)–(5.6). Then, u satisfiesEquation (5.3), namely,

(5.11)∂u

∂τ= 2λeδ/λ

(γ − p

∂u

∂y

).

Example 5.1. We take p0 = c, in which c is a constant different from zero.Equation (5.1) tells us that u0 = −c2/2. We compute γ0, δ0 and β0 by means of(4.45)–(4.47) and obtain

γ0(y, T ) = μλ,(5.12)

δ0(y, T ) = μλ(yc− λT

),(5.13)

β0(y, T ) =c2

2μexp

[μ(yc− λT

)],(5.14)

in which μ =√λc2 + λ2/λ . Equation (5.10) implies that

(5.15)

p(y, T, τ ) = c

(1 + τ (− c2

2μ + λμ+ λ)eμ(yc −λT )

)(1 + τ (− c2

2μ + λμ− λ)eμ(yc −λT )

)(−1 + τ

c2

2μeμ(

yc −λT )

)2

solves the associated Camassa-Holm equation (5.1).


Now we are ready to obtain two Darboux transforms for the associated Camassa-Holm equation (5.1):

Let us consider the quadratic pseudo-potential γ(y, T ) defined by Equations(4.45), namely,

(5.16) γy = − 1

2λpγ2 +

p

2+

λ

2p, γT =

γ2

2+

pTpγ + λu− 1

2λ2 .

The first equation in (5.16) yields

(5.17) p(y, T ) =∂

∂yγ (y, T ) +

√(∂

∂yγ (y, T )

)2

+γ (y, T )

2

λ− λ,

and replacing (5.17) into the equation for γT , we obtain a differential equation forγ. This equation is quite long, so we will not write it here explicitly. however, itcan be checked that it is invariant under the change

y → −y,√λ → −

√λ .

Equation (5.17), however, is not invariant under this change. Instead, we obtainthat

(5.18) p(y, T ) = − ∂

∂yγ (y, T ) +

√(∂

∂yγ (y, T )

)2

+γ (y, T )2

λ− λ

is also a solution to ACH. Subtracting (5.17) and (5.18) we find:

Proposition 5.2. If γ(y, T ) is determined by Equations (4.45), and p(y, T ) isa solution to the associated Camassa-Holm equation (5.1), then so is

(5.19) p(y, T ) = p(y, T )− 2∂γ

∂y(y, T ) .

Transformation (5.19) was originally found by J. Schiff in [47], using loopgroups, and an alternative derivation was presented in [27]. The foregoing deriva-tion using pseudo-potentials first appeared in the papers [25, 26] by R. HernandezHeredero and the present author.

Formulas (5.7)–(5.10) for the flow of the vector field (4.53) can be used toobtain a second Darboux transform for ACH. We recall it following [25, 26]:

Proposition 5.3. Assume that p(y, T ) solves the associated Camassa-Holmequation (5.1), and that β(y, T ) is a solution to

(5.20)βT

βy=−γ2 + λ2

p,

in which γ is a solution to the compatible system (4.45). Then, the function p(y, T )given by

(5.21) p = p

(−1 + 2τλp

−1 + τβ

(βy

p

)y

)2

− 4λ2τ2

(−1 + τβ)2β2y

p

also solves (5.1).

Proof. Start with Equation (5.10) for p(y, T ) and write it in the form

(5.22) p = p0

(−1 + ω0γ0τ

−1 + τβ0

)2

− τ2λ2

(−1 + τβ0)2p0 ω

20 .


Now use Equations (4.46) and ω0 = eδ0/λ. Then, (5.22) becomes

(5.23) p = p0

(−1 + τλp0(e

δ0/λ)y−1 + τβ0

)2

− τ2λ2

(−1 + τβ0)2p0 e

2δ0/λ .

Formula (5.21) follows from this equation by using (4.47), changing p to p, andthen dropping the subindex 0 , while Equation (5.20) is obtained directly from thesecond equation appearing in (4.47), simply using the identity eδ0/λ = 2βy/p . �

Now, the first equation in (5.16) implies that Equation (5.20) can be writtenas

βT

βy= −λ(p− 2γy),

and Proposition 5.2 says that p = p−2γy is a new solution to the ACH equation. Wetherefore interpret Proposition 5.3 as providing us with a nonlinear superpositionrule for ACH:

Corollary 5.1. Assume that p(y, T ) solves the ACH equation, and that γ(y, T )is a solution to (5.16). Set p = p− 2 γy, and assume that β(y, T ) satisfies βT /βy =−λ p. Then, p is a solution to ACH and so is p given by (5.21).

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Departamento de Matematica y Ciencia de la Computacion, Universidad de Santia-

go de Chile, Casilla 307 Correo 2, Santiago, Chile.



Explicit higher-dimensional Darboux transformations for thetime-dependent Schrodinger equation

Axel Schulze-Halberg

Abstract. We construct Darboux transformations for time-dependentSchrodinger equations in arbitrary spatial dimensions. The Darboux operatorthat connects a pair of Schrodinger equations and the corresponding potentialdifference are obtained in explicit form. An example in (5 + 1) dimensions ispresented and the representation of the Darboux operator in different coodi-nate systems is discussed.

Contents

Introduction1. Preliminaries2. The Darboux operator in (n+ 1) dimensions3. Application: the free particle in (5 + 1) dimensions4. Darboux transformations in arbitrary coordinate systemsConcluding remarksReferences

Introduction

The Darboux transformation is a mathematical construction that allows for thegeneration of solutions to certain differential equations. A pair of such equationsis said to admit a Darboux transformation, if their solutions can be transformedinto each other by applying a linear differential operator, called Darboux operator.While the Darboux transformation was first introduced for ordinary, second-orderequations only [8], in the meantime it has been generalized to become a popularmethod for generating solutions of many linear and nonlinear equations, such asthe Dirac equation, linear and nonlinear Schrodinger equations, the Korteweg-de-Vries equation, the sine-Gordon equation and many more, see [10] and [15] for anoverview. Especially in Quantum Physics the Darboux transformation has becomefamous, as it represents the mathematical core of the supersymmetry (SUSY) for-malism [2]. In the context of this SUSY formalism, the Darboux transformationhas been applied to the Schrodinger equation and has generated a large number

2000 Mathematics Subject Classification. Primary 81Q05; Secondary 81Q60, 81Q80.Key words and phrases. Time-dependent Schrodinger equation, Darboux transformation,

potential difference.


165

166 A. SCHULZE-HALBERG

of results regarding solvable potentials and spectral design, in particular for theSchrodinger equation, see the reviews [7] [13] for details. The Darboux trans-formation for the Schrodinger equation has been thoroughly studied regarding itsmathematical properties, such as factorizability of the Darboux operator [1], appli-cability to generalized Schrodinger equations [19], behaviour of Green’s functionsfor Darboux-transformed equations [18], just to name a few. These properties,however, as well as the above mentioned applications have been established in onespatial dimension only. This includes higher-dimensional cases that allow for re-duction to one dimension, such as by separation of variables or using symmetriesin the equation [12] [5] [3]. In fact, there are very few studies regarding Darbouxtransformations in higher spatial dimensions. An example is the analysis carriedout in [14] and its subsequent application [9], where the multidimensional SUSYformalism for the stationary Schrodinger equation is developed and applied in thetwo-dimensional case for the construction of isospectral potential families. Whilethe latter results concern the stationary Schrodinger equation, in this note we willconsider the fully time-dependent case. Based on our previous work [17], wherewe evaluated the n-dimensional intertwining relation and obtained a correspond-ing Darboux transformation in the special case of two spatial dimensions, in thepresent work we will construct the explicit Darboux operator in arbitrary spatialdimensions. In section 2 we give a brief summary of the one-dimensional Darbouxtransformation and we review results from [17]. The construction of the Darbouxoperator is done in section 3, followed by an application (section 4). The finalsection 5 is devoted to the representation of our Darboux operator and the corre-sponding potential difference in different coordinate systems, an issue that does notexist in one dimension.

1. Preliminaries

We will now review the concept of a Darboux transformation and summarizesome results from [17], which will be needed in the subsequent sections. Considera Schrodinger equation in atomic units (� = 1, mass 1/2):

i Ψt +�Ψ− V1 Ψ = 0,(1)

where the index denotes partial differentiation, � = ∇2 stands for the Laplacianwith respect to the spatial variables x1, ..., xn, the function Ψ = Ψ(x1, ..., xn, t)is a solution and V1 = V1(x1, ..., xn, t) denotes the potential. We want to relatesolutions Ψ of (1) to solutions Φ = Φ(x1, ..., xn, t) of a second Schrodinger equation

i Φt +�Φ− V2 Φ = 0,(2)

with potential V2 = V2(x1, ..., xn, t) via the first-order differential operator

Dn = L0 +

n∑j=1

Lj ∂xj

= L0 + L ∇,(3)

where ∂ stands for a partial derivative operator, L = (L1, ..., Ln)T and Lj =

Lj(x1, ..., xn, t), j = 1, ..., n, are such that (3) maps solutions of (1) onto solutionsof (2), that is,

Dn(Ψ) = Φ.

EXPLICIT HIGHER-DIMENSIONAL DARBOUX TRANSFORMATIONS 167

If this property holds, then Dn is called Darboux operator for the Schrodingerequations (1) and (2), and the application of Dn to a solution of (1) is calledDarboux transformation. Note that there is no derivative with respect to t in (3),as this would force the difference between the potentials V1 and V2 to vanish, ascan be seen as follows. It is well-known [17] that in the intertwining relation usedfor the construction of our Darboux operator,

(i ∂t +�− V2 )D = D (i ∂t +�− V1) ,(4)

the coefficient of the time derivative operator ∂t vanishes. Let us now suppose thatour Darboux operator (3) contained a derivative with respect to t. According to (4),intertwining of the latter derivative with the Schrodinger operator correspondingto our transformed equation (2) gives

(i ∂t +�− V2 ) ∂t = i ∂tt +� ∂t − V2 ∂t,(5)

while the reverse process applied to our initial equation (1) yields

∂t (i ∂t +�− V1 ) = i ∂tt + ∂t �− (V1)t − V1 ∂t,(6)

For sufficiently smooth arguments ∂t and � will commute, such that comparisonof (5) and (6) gives the coefficient V2 − V1 of ∂t. This coefficient only vanishes ifV1 = V2, since we know that the conventional intertwining relation (4) does notcontain any terms related to ∂t. Now, for the Darboux operator (3) to exist, aninterrelation between the potentials V1 and V2 must be imposed. Furthermore,the coefficient functions Lj , j = 0, ..., n must meet certain conditions [17] that wesummarize in the following points.

• Our coefficient functions Lj , j = 1, ..., n, written as components of thevector L, have the following explicit form:

L =

⎛⎜⎜⎜⎜⎜⎝L1

L2

L3

...Ln

⎞⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝l10l20l30...ln0

⎞⎟⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎜⎝0 l12 l13 · · · l1n

−l12 0 l23 · · · l2n−l13 −l23 0 · · · l3n...

......

. . ....

−l1n −l2n −l3n · · · 0

⎞⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝x1

x2

x3

...xn

⎞⎟⎟⎟⎟⎟⎠ ,(7)

for arbitrary matrix entries lj0 = lj0(t), j = 1, ..., n and ljk = ljk(t),j, k = 1, ..., n, where these functions are assumed to be continuouslydifferentiable. Note that (7) reduces correctly to the well-known one-dimensional case for n = 1, as we get L = L1 = l10. For more details onthis case, the reader may consult [1] and references therein.• The coefficient function L0 is related to the to the potential differenceV1 − V2 and to the other coefficient functions Lj , j = 1, ..., n via

∇L0 = −1

2

[i Lt + (V1 − V2) L

],(8)

which is a vectorial equation with n components. This equation reducescorrectly to its one-dimensional counterpart [1], if n is set to one.• The function L0 must fulfill the following equation

i (L0)t +�L0 + (V1 − V2) L0 + L (∇V1) = 0,(9)

which we will refer to as auxiliary equation. For n = 1 the well-knownone-dimensional equation [1] is recovered from (9).


Thus, in order to construct a Darboux operator (3) for our Schrodinger equations(1) and (2), we need to solve the n+ 1 equations in (8) and (9).

2. The Darboux operator in (n + 1) dimensions

The first part of this section is devoted to constructing an interrelation betweenthe potentials V1 and V2, associated with the Schrodinger equations (1) and (2),respectively. In order to do so, we will also have to put certain restrictions onthe entries of (7). In the second part we will find the function L0, which togetherwith (7) determines the Darboux operator (3). In the final part of this sectionwe will study the auxiliary equation (9) that must be fulfilled for the Darbouxtransformation to exist.

2.1. The potential difference. The potential difference is a central term inthe one-dimensional formalism of Darboux transformations. The Darboux operatoronly exists if the potential difference meets a certain form that can be expressedthrough the function L0. This form is easily obtained from equation (8) by solvingfor the potential difference, recall that in one dimension we have L1 = l10 and thatx1 is the only spatial variable [1]:

V1 − V2 = −i l′10l10− 2 (L0)x1

.(10)

While expressing the potential difference through L0 is simple in one dimension, inthe general, higher-dimensional case existence of L0 is not guaranteed. Note that(8) represents a constraint on the gradient of L0, and as such does not necessarilyhave a solution for L0, except if an integrability condition is fulfilled:

(L0)xjxk= (L0)xkxj

, j, k = 1, ..., n

[i (Lj)t + (V1 − V2) Lj ]xk= [i (Lk)t + (V1 − V2) Lk]xj

, j, k = 1, ..., n.(11)

Due to symmetry it is no restriction to assume that the indices j, k belonging to acomponent of (11) satisfy j < k, which reduces the number of different componentsto n(n − 1)/2. Note that the case of (2 + 1) dimensions is particularly simple,as the integrability condition (11) then has only one component. In general, weobserve that (11) can only be fulfilled by choosing the potential difference V1 − V2

appropriately, as it is the only free, nonconstant parameter in (11). Now considerthe following potential difference:

V1 − V2 = −i l′12l12

+1

L22

u,(12)

where u is an arbitrary function that depends on its two arguments as follows:

u = u

(t,L1

L2

).(13)

In the following we will show that the function (12), (13) solves our integrabilitycondition (11). Before we do so, let us mention that the form of the potentialdifference (12) was chosen in accordance with the two-dimensional case [17], as wewere unable to access any other more general solution. Since our function u in (13)depends on the two spatial variables x1, x2 only, all components of our condition(11) that satisfy j, k > 2 will be easily satisfied (by appropriately choosing thecoefficients ljk, j, k = 1, ..., n), as the partial derivatives of the potential differencevanishes. Consequently, one can choose a different pair of spatial variables xj , xk,


j, k = 1, ..., n and their corresponding functions Lj , Lk, j, k = 1, ..., n in (12) and(13). This will not change the structure of the transformed potential, but onlyits dependence on the variables. For the sake of simplicity, we will focus here onthe scenario of the first two spatial variables. It is important to notice that udepends on only one argument that contains spatial variables. As a consequence,the potential difference (12) depends on two arguments that involve these spatialvariables, namely, L1/L2 and L2. We will come back to this issue in our applica-tion section 4. Next, we show that the potential difference (12), (13) solves ourintegrability condition (11) for all j and k. For this to be true, we need to restrictthe free parameters in our coefficient functions Lj , j = 1, ..., n, as given in (7). Ourvector L = (L1, ..., Ln)

T then receives the following form, introducing constantspjk, 1 ≤ j < k ≤ n:

L =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

l10l20l30l40...ln0

⎞⎟⎟⎟⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 p12 l12 p13 l12 p14 l12 · · · p1n l12−p12 l12 0 p23 l12 p24 l12 · · · p2n l12−p13 l12 −p23 l12 0 p34 l12 · · · p3n l12−p14 l12 −p24 l12 −p34 l12 0 · · · p4n l12...

......

.... . .

...−p1n l12 −p2n l12 −p3n l12 −p4n l12 · · · 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝

x1

x2

x3

x4

...xn

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

(14)

The functions lj0 = lj0(t), j = 1, ..., n, must be constrained as follows:

lj0 = p1j l20 − p2j l10, j = 3, ..., n.(15)

In particular, the functions l10 and l20 stay arbitrary. Furthermore, the constantspjk, 1 ≤ j < k ≤ n, are required to fulfill the following additional interrelations:

p12 = 1(16)

pjk = p1j p2k − p1k p2j , 3 ≤ j < k ≤ n.(17)

Note that the constants pjk for j = 1, 2 and k = 3, ..., n stay arbitrary. Now we areready to show that our potential difference (12), together with the settings (14)-(17) satisfies our integrability condition (11). For the sake of simplicity, we will usethe name ljk when referring to the entries of the square matrix in L, although theyhave been redefined in terms of l12 and the constants pjk. Now fix a j and a kwith 1 ≤ j < k ≤ n and evaluate the corresponding component of our integrabilitycondition (11):

[i(Lj)t]xk+ [(V1 − V2) Lj ]xk

= [i (Lk)t]xj+ [(V1 − V2) Lk]xj

i l′jk +

[−i l′12

l12Lj +

Lj

L22

u

]xk

= i l′kj +

[−i l′12

l12Lk +

Lk

L22

u

]xj

i l′jk − il′12l12

ljk +

[Lj

L22

u

]xk

= i l′kj − il′12l12

lkj +

[Lk

L22

u

]xj

2 i

(l′jk −

l′12l12

ljk

)=

[Lk

L22

u

]xj

−[Lj

L22

u

]xk

,(18)


where in the last step we used the symmetry lkj = −ljk, j, k = 1, ..., n. Let us firstevaluate the right hand side of this equation. We get[

Lk

L22

u

]xj

−[Lj

L22

u

]xk

=

=

[(Lk)xj

L22 − 2 Lk L2 (L2)xj

L42

− (Lj)xkL22 − 2 Lj L2 (L2)xk

L42

]u+(19)

+

[Lk (L1)xj

L2 − Lk L1 (L2)xj

L42

− Lj (L1)xkL2 − Lj L1 (L2)xk

L42

](u)2(20)

Here, (u)2 stands for the partial derivative of u with respect to its second argument.We show that the coefficients of u and of its derivative are zero. Starting with thederivative, we have to distinguish several cases regarding the values of j and k.

• Assume j = 1 and k = 2.

Lk (L1)xjL2 − Lk L1 (L2)xj

− Lj (L1)xkL2 + Lj L1 (L2)xk

=

= L2 ll1 L2 − L2 L1 l21 − L1 l12 L2 + L21 l22

= −L1 L2 l21 − L1 L2 l12

= 0,

where we used the symmetry l12 = −l21 and l11 = l22 = 0.• Assume j = 1 and k ≥ 3.

Lk (Ll)xjLm − Lk Ll (Lm)xj

− Lj (Ll)xkLm + Lj Ll (Lm)xk

=

= −Lk L1 (L2)x1− L1 (L1)xk

L2 + L1 L1 (L2)xk

= −L1 Lk l21 − L1 L2 l1k + L21 l2k

= L1 (−Lk l21 − L2 l1k + L1 l2k).(21)

We will now prove that the term in brackets vanishes. To this end, weneed to use its explicit form that we infer from the representation of Lgiven in (14)-(17). We have

Lk l21 =

⎡⎢⎣p1k l20 − p2k l10 − p1k l12 x1 − p2k l12 x2 −n∑

q=3q �=k

(p1q p2k − p1k p2q) l12 xq

⎤⎥⎦ l21.

(22)

The remaining two terms in brackets on the right hand side of (21) aregiven by

(23)

−L2 l1k+L1 l2k =

(l20 − l12 x1 +

n∑q=3

p2q l12 xq

)p1k−

(l10 +

n∑q=3

p1q l12 xq

)p2k.

It is immediate to see that the right hand sides of (22) and (23) are thesame. This implies that their difference vanishes, which shows

L1 (−Lk l21 − L2 l1k + L1 l2k) = 0.(24)


• Assume j = 2 and k ≥ 3.



=

= Lk (L1)x2L2 − Lk L1 (L2)x2

− L2 (L1)xkL2 + L2 L1 (L2)xk

= L2 Lk l12 − L22 l1k + L1 L2 l2k

= L2 l12 (Lk − L2 p1k + L1 p2k)

= 0,

where in the last step we used (24), together with the symmetry l21 =−l12.

• Assume 3 ≤ j < k.



=

= Lk (L1)xjL2 − Lk L1 (L2)xj

− Lj (L1)xkL2 + Lj L1 (L2)xk

= Lk (L2 l1j − L1 l2j)− Lj (L2 l1k − L1 l2k)

= −Lk Lj + Lj Lk

= 0.

In the second last step we used (24).

We proceed with the remaining term (19). This term is proportional to u and thenumerator of its coefficient must be shown to vanish for all indices 1 ≤ j < k ≤ n.

• Assume j = 1 and k = 2.

(Lk)xjL22 − 2 Lk L2 (L2)xj

− (Lj)xkL22 + 2 Lj L2 (L2)xk

=

= (L2)x1L22 − 2 L2 L2 (L2)x1

− (L1)x2L22 + 2 L1 L2 (L2)x2

= L22 l21 − 2 L2

2 l21 − L22 l12 + 2 L1 L2 l22

= L22 l21 − 2 L2

2 l21 − L22 l12

= 0,

because we have l21 = −l12.• Assume j = 1 and k ≥ 3.


− (Lj)xkL22 + 2 Lj L2 (L2)xk

=

= (Lk)x1L22 − 2 Lk L2 (L2)x1

− (L1)xkL22 + 2 L1 L2 (L2)xk

= L22 lk1 − 2 L2 Lk l21 − L2

2 l1k + 2 L1 L2 l2k

= 2 L2 (−L2 l1k − Lk l21 + L1 l2k)

= 0.

In the second last step we used l1k = −lk1, and in the last step (24) wasemployed.• Assume j = 2 and k ≥ 3.


− (Lj)xkL22 + 2 Lj L2 (L2)xk

=

= (Lk)x2L22 − 2 Lk L2 (L2)x2

− (L2)xkL22 + 2 L2 L2 (L2)xk

= L22 lk2 − 2 L2 Lk l22 − L2

2 l2k + 2 L22 l2k

= L22 (lk2 − l2k + 2 l2k)

= 0,

due to the symmetry l2k = −lk2.


• Assume 3 ≤ j < k.


− (Lj)xkL22 + 2 Lj L2 (L2)xk

=

= L22 lkj − 2 L2 Lk l2j − L2

2 ljk + 2 L2 Lj l2k

= 2 L2 (L2 lkj − Lk l2j + Lj l2k)

= 2 L2 l12 [L2 (p1k p2j − p1j p2k)− Lk p2j + Lj p2k].(25)

We will now prove that the term in square brackets is zero. To this end,we insert the explicit form of L resp. its components, as given in (14)-(17):

L2 (p1kp2j − p1j p2k) =

=(l20 − l21 x1 +

n∑q=3

p2q l12 xq

)(p1k p2j − p1j p2k)

= p1k p2j l20 − p1j p2k l20 − p1k p2j l21 x1 + p1j p2k l21 x1+

+ p1k p2j

n∑q=3

p2q l12 xq − p1j p2k

n∑q=3

p2q l12 xq(26)

The remaining terms in square brackets on the right hand side of (25)read

− Lk p2j + Lj p2k =

= −[p1k l20 − p2k l10 − pk1 l12 x1 − pk2 l12 x2 −

n∑q=3

(p1q p2k − p1k p2q) l12 xq

]p2j+

+[p1j l20 − p2j l10 − pj1 l12 x1 − pj2 l12 x2 −

n∑q=3

(p1q p2j − p1j p2q) l12 xq

]p2k

= −p1k p2j l20 + p1j p2k l20+

+ (p2j pk1 l12 − pj1 p2k l12) x1 + (p2j pk2 l12 − pj2 p2k l12) x2−

−n∑

q=3

(p1q p2j p2k − p2j p1k p2q) l12 xq −n∑

q=3

(p1q p2j p2k − p1j p2k p2q) l12 xq.

(27)

By comparison of (26) and (27) it is easy to see that their sum yieldszero, as any term in (26) appears in (27), but with opposite sign. Thisimplies that (25) vanishes.

We have shown that (20) and (19) both vanish. It remains to consider the left handside of our integrability condition (18), which must be shown to equal zero. To thisend, we first observe that all entries ljk, j, k = 1, ..., n of the square matrix in (14)are proportional to l12 (this is true even for the diagonal, where l12 can be seenas multiplied by zero), say, we have ljk = ajk l12 for appropriate constants ajk,j, k = 1, ..., n. Now we substitute this form of ljk into the left hand side of (18):

l′jk −l′12l12

ljk = ajk l′12 −l′12l12

ajk l12

= 0.

So both sides of equation (18) vanish, implying that our potential difference (12),(13) solves all components of the integrability condition (11), if the settings (14)-(17) are employed. Before we proceed, let us point out that for n = 1 the potential


difference (12) becomes undefined and, in particular, does not reduce to its one-dimensional counterpart (10). This is due to the fact that the potential differenceis not obtained by solving (8) directly, but from the integrability condition thatdoes not exist in one dimension. Another important comment concerns the realityof our transformed potential V2, as given in (12): similar to the one-dimensionalcase [1], the transformed potential can become complex-valued, depending on theinitial potential V1 and the function u. In some cases, one can use the first term onthe right-hand side of (12) to absorb imaginary expressions (reality condition). Ageneral statement or condition for the reality of the transformed potential cannotbe given, as too many parameters are involved. As far as the physical meaning ofour transformed potential is concerned, usually complex-valued functions have lessapplications that their real counterparts.

2.2. Construction of the function L0. Since the integrability condition(11) has been solved, equation (8) is guaranteed to admit a solution for L0. Similarto the previous paragraph, we will now present an explicit function L0 and show thatit solves equation (8). To this end, let us first define a function v that depends on thesame arguments as u, see (13), such that u = (v)2, where the index represents thepartial derivative with respect to the second argument. Now consider the followingfunction:

L0 = − 1

2 l12v − i

2

n∑q=1

(l′q0 −

lq0 l′12l12

)xq,(28)

where the functions lq0, q = 1, ..., n, are defined in (15). Suppose that the poten-tial difference is chosen according to (12), (13), and substitute (28) into the j-thcomponent of equation (8). Its left hand side then becomes

(L0)xj=

[− 1

2 l12v − i

2

n∑q=1

(l′q0 −

lq0 l′12l12

)xq

]xj

= − 1

2 l12vxj− i

2

(l′j0 −

lj0 l′122 l12

)= − 1

2 l12

(L1

L2

)xj

(v)2 −i

2

(l′j0 −

lj0 l′122 l12

)= −

(L1)xjL2 − L1 (L2)xj

2 l12 L22

u− i

2

(l′j0 −

lj0 l′122 l12

)= −p1j l12 L2 − L1 p2j l12

2 l12 L22

u− i

2

(l′j0 −

lj0 l′122 l12

)= − Lj

2 L22

u− i

2

(l′j0 −

lj0 l′122 l12

).(29)

Note that in the last step we made use of the identity (24). Next, we substitutethe explicit form of our potential difference (12) and of Lj for a fixed j - see (14) -


into the right hand side of equation (8). This gives

−1

2

[i (Lj)t + (V1 − V2) Lj

]=

= −1

2

⎡⎢⎣i l′j0 + i

n∑q=1q �=j

pjq l′12 xq +

(−i

l′12l12

+1

L22

u

)⎛⎜⎝lj0 +

n∑q=1q �=j

pjq l12 xq

⎞⎟⎠⎤⎥⎦

= −1

2

⎡⎢⎣i l′j0 + i

n∑q=1q �=j

pjq l′12 xq − il′12l12

⎛⎜⎝lj0 +

n∑q=1q �=j

pjq l12 xq

⎞⎟⎠+Lj

L22

u

⎤⎥⎦= −1

2

⎡⎢⎣i l′j0 − ilj0 l′12l12

+ in∑

q=1q �=j

pjq l′12 xq − il′12l12

⎛⎜⎝ n∑q=1q �=j

pjq l12 xq

⎞⎟⎠+Lj

L22

u

⎤⎥⎦= −1

2

[i l′j0 − i

lj0 l′12l12

+Lj

L22

u

].(30)

Comparison of (29) and (30) gives equivalence. Since the choice of j was arbitrary,we obtain that L0 as given in (28) fulfills all components of equation (8).

2.3. The auxiliary equation. Now that we have solved (8), the only remain-ing condition to be satisfied is the auxiliary equation (9). In the one-dimensionalcase, at this point the function L0 is still arbitrary and can be chosen in such away that the auxiliary equation linearizes to a Schrodinger equation. In the presentcase of higher dimensions, this approach does not work, simply because L0 is notfree anymore, but has already been determined as shown in (28). Therefore, theauxiliary equation (9) cannot be seen as an equation for L0, but must be solved byemploying a different function that has not been determined so far. Once L0, L andthe potential difference V1 − V2 have been substituted into the auxiliary equation(9), the only arbitrary functions that remain in this equation are v from (28) andthe initial potential V1. As for the first of these two functions, in general v cannotbe used to solve the auxiliary equation (9) for any initial potential V1, simply be-cause this potential can depend arbitrarily on all variables, while v (as its derivativeu) has only two arguments, as shown in (12). This means that only for very fewand particular choices of V1 the auxiliary equation becomes solvable through v. Ingeneral, only the initial potential V1 remains unrestricted, such that the auxiliaryequation (9) can always be solved with respect to V1, provided the solution takesa closed form. An immediate consequence of this fact is that in dimensions higherthan one, the Darboux transformation cannot be applied to any initial potential,but only to the particular class of functions V1 that solve the auxiliary equation(9). This is a strong contrast to the one-dimensional case, where both functions L0

and V1 remain arbitrary in the auxiliary equation, such that L0 can be expressedthrough the function V1, which stays free. Mathematically, the auxiliary equation(9) is a linear, first-order partial differential equation with respect to V1 with poly-nomial coefficients Lj , j = 1, ..., n. Solutions of such equations can be given inclosed form, but they usually involve an integral that cannot be resolved until thefunction v has been determined. It is therefore in general difficult to check whethera particular potential can be transformed using our Darboux operator.


3. Application: the free particle in (5 + 1) dimensions

In this section we will apply our Darboux transformation to the free particleSchrodinger equation in five spatial dimensions. To this end, consider equation (1)for a free particle, that is, for the potential V1 = 0:

i Ψt +�Ψ = 0,(31)

where we will work in (5 + 1) dimensions, that is, the solution Ψ depends on thevariables x1, ..., x5 and t. In order to perform our Darboux transformation, we mustconstruct the Darboux operator (3), which reads in the present case n = 5:

D5 = L0 + L1∂

∂x1+ L2

∂

∂x2+ L3

∂

∂x3+ L4

∂

∂x4+ L5

∂

∂x5

= L0 + L ∇,(32)

where as usual L stands for the vector L = (L1, ..., L5)T . Before we deal with the

first coefficient L0, we construct the vector L, determined by (14)-(17). For thesake of simplicity we make the following settings:

l10 = 2 t+ 2 l20 = t+ 1 l12 = 1(33)

p13 = 2 p14 = 3 p15 = 4 p23 = 1 p24 = 2 p25 = 3.(34)

It is straightforward to verify that on plugging the above settings into (14)-(17),we get

L =

⎛⎜⎜⎜⎜⎝2 t+ 2t+ 10−t− 1−2 t− 2

⎞⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎝0 1 2 3 4−1 0 1 2 3−2 −1 0 1 2−3 −2 −1 0 1−4 −3 −2 −1 0

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

x1

x2

x3

x4

x5

⎞⎟⎟⎟⎟⎠ .(35)

In the next step we construct the first coefficient L0 in our Darboux operator (32),the general form of which can be extracted from (28). In the present case we find,after taking into account n = 5 and our settings (33)-(34), the following functionL0:

L0 = −1

2v − i

(x1 +

x2

2− x4

2− x5

),(36)

where the function v depends on the same arguments as u in (13). Now, our lastremaining task is to satisfy the auxiliary equation (9). To this end, we substituteour functions L0, ..., L5, as given in (35) and (36), respectively, and the potentialdifference V1 − V2 as given in (12):

V1 − V2 = −V2 =

(1

t+ 1− x1 + x3 + 2 x4 + 3 x5

)2

(v)2.(37)

Furthermore, for the sake of simplicity we will now make the restriction that vdepends only on its second argument, which means in the present case that

v = v

(L1

L2

)= v

(2 t+ 2 + x2 + 2 x3 + 3 x4 + 4 x5

t+ 1− x1 + x3 + 2 x4 + 3 x5

).(38)

It is immediate to see that our transformed potential V2 depends only on twodifferent arguments that involve spatial variables, namely L1/L2 and L2. As aconsequence, the solutions of our transformed equation (2) will also depend onthese two arguments only. Let us point out that this is not a general property


of our Darboux transformation, but due to the fact that V1 vanishes. Now, aftersubstitution of the above functions and after abbreviating the argument of v in (38)as z, the auxiliary equation (9) takes the following form:

(15 z2 − 40 z + 30) v′′ + (30 z − 40) v′ + v′ v = 0,(39)

where the prime denotes the derivative with respect to z, recall that due to (38), thefunction v depends only on z. A solution of equation (39) is given by the followingfunction:

v =√2 tanh

[1

10arctan

(3 z − 4√

2

)].(40)

Now we have determined our Darboux operator (32) completely. Its explicit formis obtained by inserting (35), (36) and (40):

D5 = −√

1

2tanh

{1

10arctan

[2 t+ 2 + 4 x1 + 3 x2 + 2 x3 + x4√2 (t+ 1− x1 + x3 + 2 x4 + 3 x5)

]}−

− i

(x1 +

x2

2− x4

2− x5

)+

+

(2 t+ 2 + x2 + 2 x3 + 3 x4 + 4 x5

)∂

∂x1+

+

(t+ 1− x1 + x3 + 2 x4 + 3 x5

)∂

∂x2+

+

(− 2 x1 − x2 + x4 + 2 x5

)∂

∂x3+

+

(− t− 1− 3 x1 − 2 x2 − x3 + x5

)∂

∂x4+

+

(− 2 t− 2− 4 x1 − 3 x2 − 2 x3 − x4

)∂

∂x5.(41)

This is the Darboux operator in (5+1) dimensions for the free-particle Schrodingerequation (31) and its transformed counterpart (2). If we apply (41) to a solution ofthe Schrodinger equation (31), then we get a solution of the Schrodinger equation(2), where the potential V2 is obtained from (37) in combination with (40). Recallingthat z is given by the argument of v in (38), we arrive at

V2 = − 3

10 (t+ 1− x1 + x3 + 2 x4 + 3 x5)2 + 5 (2 t+ 2 + 4 x1 + 3 x2 + 2 x3 + x4)2×

× cosh−2

{1

10arctan

[2 t+ 2 + 4 x1 + 3 x2 + 2 x3 + x4√2 (t+ 1− x1 + x3 + 2 x4 + 3 x5)

]}.(42)

Before we conclude this section, let us employ the Darboux operator (41) to gen-erate an explicit solution of the Schrodinger equation (2) with potential (42). Tothis end, we need a solution of our initial, free-particle Schrodinger equation (31),which we choose as

Ψ = exp

[− i

5

(t+ x1 + x2 + x3 + x4 + x5

)].(43)


It is immediate to see that this function solves equation (31). Now, on applyingthe Darboux operator (41) to the solution (43), we obtain the following result:

D5(Ψ) =

(i√2 tanh

{1

10arctan

[2 t+ 2 + 4 x1 + 3 x2 + 2 x3 + x4√2 (t+ 1− x1 + x3 + 2 x4 + 3 x5)

]}+

+ 2 x1 + x2 − x4 − 2 x5

)exp

[− i

5

(t+ x1 + x2 + x3 + x4 + x5

)].

This functionD5(Ψ) = Φ is an explicit solution of the (5+1)-dimensional Schrodingerequation (2) for the potential V2 as given in (42).

4. Darboux transformations in arbitrary coordinate systems

In higher spatial dimensions it is common to rewrite the Schrodinger equation incoordinates different from the usual cartesian ones, e.g. if the boundary conditionsor the underlying potential possess a certain symmetry. Clearly, such a change ofvariables in the Schrodinger equation extends to the Darboux transformation thatwe have constructed here. As the general procedure of switching coordinates isstraightforward, we consider it sufficient to present our Darboux transformation intwo particular coordinate systems.

Polar coordinates in (2+1) dimensions. Introduction of polar coordinates intwo spatial dimensions means to change coordinates from x1, x2 to y1, y2 via therelations x1 = y1 cos(y2) and x2 = y1 sin(y2), where y1 > 0 and 0 ≤ y2 < 2π. Incartesian coordinates, our Darboux operator D2 can be obtained from (14)-(17)and from (28):

D2 = L0 + L ∇

= L0 + L1∂

∂x1+ L2

∂

∂x2,(44)

where the coefficient functions are given by

L0 = − 1

2 l12v − i

2

(l′10 −

l10 l′12l12

)x1 −

i

2

(l′20 −

l20 l′12l12

)x2

L1 = l10 + l12 x2

L2 = l20 − l12 x1.

Note that the function v depends on the two variables t and L1/L2. On switchingto polar coordinates, the spatial variables and derivatives in our Darboux operator(44) change according to the definition of polar coordinates and the chain rule,respectively. We obtain the following form for (44):

D2 = L0 + L1∂

∂y1+ L2

∂

∂y2,


where the coefficients change after substituting the new coordinates y1, y2 and afterrewriting the derivatives as follows:

L0 = − 1

2 l12v − i

2

(l′10 −

l10 l′12l12

)y1 cos(y2)−

i

2

(l′20 −

l20 l′12l12

)y1 sin(y2)

L1 = −l10 cos(y2)− l20 sin(y2)

L2 = −l12 −l20 cos(y2)

y1+

l10 sin(y2)

y1.

(45)

Note that the dependency of v on its variables now reads as follows:

(46) v = v

(t,l10 − l12 y1 sin(y2)

l20 + l12 y1 cos(y2)

).

Two Schrodinger equations in polar coordinates that are connected via our Darbouxoperator (45), exhibit the following explicit potential difference (12):

V1 − V2 = −i l′12l12

+

(1

l20 + l12 y1 cos(y2)

)2

u,

where u = (v)2 depends on the same argument as v in (46).Parabolic coordinates in (3+1) dimensions. While parabolic coordinates are

usually used in two dimensions, there is a less common three-dimensional version,defined by the following relations:

x1 = y1 y2 cos(y3)

x2 = y1 y2 sin(y3)

x3 =y212− y22

2,

where y1, y2 > 0 and 0 ≤ y3 < 2π. Before we express our Darboux operator inthese coordinates, let us extract its cartesian form from (14)-(17) and from (28):

D2 = L0 + L ∇

= L0 + L1∂

∂x1+ L2

∂

∂x2+ L3

∂

∂x3.(47)

Here the coefficient functions have the following explicit form:

L0 = − 1

2 l12v − i

2

(l′10 −

l10 l′12l12

)x1 −

i

2

(l′20 −

l20 l′12l12

)x2 +

+i

2

(p23 l′10 − p13 l′20 −

p23 l10 l′12l12

+p13 l20 l′12

l12

)x3

L1 = l10 + l12 x2 + p13 l12 x3

L2 = l20 − l12 x1 + p23 l12 x3

L3 = p13 l20 − p23 l10 − p13 l12 x1 − p23 l12 x2,


where the function v is allowed to depend on the two variables t and L1/L2. Afterchanging to parabolic coordinates, the Darboux operator (47) takes the followingform:

D2 = L0 + L1∂

∂y1+ L2

∂

∂y2+ L3

∂

∂y3.

The coefficients are now expressed through parabolic coordinates and the deriva-tives in (47) have been adjusted:

L0 = − 1

2 l12v − i

2

(l′10 −

l10 l′12l12

)y1 y2 cos(y3)−

i

2

(l′20 −

l20 l′12l12

)y1 y2 sin(y3)+

+i

4

(p23 l′10 − p13 l′20 −

p23 l10 l′12l12

+p13 l20 l′12

l12

) (y21 − y2

2

)

L1 =(p23 l10 − p13 l20) y1

y21 + y2

2

+

(p13 l12 y2

2+

l10 y2y21 + y2

2

)cos(y3)+

+

(p23 l12 y2

2+

l20 y2y21 + y2

2

)sin(y3)

(48)

L2 = − (p23 l10 + p13 l20) y2y21 + y2

2

−(

l10 y1y21 + y2

2

+p13 l12 y1 y2 (y2

1 + y2)

2 (y21 + y2

2)

)cos(y3)+

+

(−p23 l12 y1

2+

l20 y1y21 + y2

2

)sin(y3)

(49)

L3 = −l12 +

(−p23 l12 y1

2 y2+

p23 l12 y22 y1

+l20

y1 y2

)cos(y3)+

+

(−p13 l12 y2

2 y1+

p13 l12 y12 y2

− l10y1 y2

)sin(y3).

The function v that appears in L0 depends on its arguments in the followingexplicit way:

v = v

(t,2 l10 − p13 l12 y21 + p13 l12 y22 + 2 l12 y1 y2 sin(y3)

2 l20 − p13 l12 y22 + p13 l12 y21 − 2 l12 y1 y2 cos(y3)

).

The potential difference is given by (12) with L1 and L2 being substituted by thefunctions (48), (49) given above. Due to the length of the resulting expression, weomit to display the explicit form of the potential difference.

Concluding remarks

We have presented an approach to a generalization of the Darboux transforma-tion for the time-dependent Schrodinger equation to arbitrary spatial dimensions.The main difference to the well-known one-dimensional case lies in the fact thatthe higher-dimensional Darboux transformation cannot be applied to any potential,but only to a particular class of potentials. Despite this restriction, recent resultsshow that in the stationary, two-dimensional case chains of Darboux transforma-tions can be constructed [16], and that generalized Schrodinger equations, such asthe effective mass case, also admit Darboux transformations [6]. However, at thispoint it is not clear whether and how the results presented in this work and theabove given references relate to findings on the two-dimensional SUSY formalism


and shape-invariant potentials, see [4] [11] and references therein. This aspect issubject of currently ongoing research.

References

[1] V.G. Bagrov and B.F. Samsonov, “Darboux transformation, factorization, supersymmetryin one-dimensional quantum mechanics”, Theor. Math. Phys. (1995), 1051-1060 MR1488681(98m:81042)

[2] V.G. Bagrov and B.F. Samsonov, “Supersymmetry of a nonstationary Schrodinger equation”,Phys. Lett. A 210 (1996), 60-64 MR1372678 (96m:81067)

[3] G.G. Blado, “Supersymmetric treatment of a particle subjected to a ring-shaped potential”,Int. J. Quant. Chem. 58 (1998), 431 - 439

[4] F. Cannata, M.V. Ioffe and D.N. Nishnianidze, “New two-dimensional quantum models withshape invariance”, J. Math. Phys. 52 (2011), 022106 (9pp) MR2798376

[5] F. Cannata, M.V. Ioffe, and D.N. Nishnianidze, “New methods for the two-dimensionalSchrodinger equation: SUSY-separation of variables and shape invariance”, J. Phys. A 35(2000), 1389-1404 MR1890922 (2003e:81202)

[6] H. Cobian and A. Schulze-Halberg, “Time-dependent Schodinger equations with effectivemass in (2+1) dimensions: intertwining relations and Darboux operators”, J. Phys. A 44(2011), 285301 MR2812346

[7] F. Cooper, A Khare and U. Sukhatme, “Supersymmetry and Quantum Mechanics”, Phys.Rep. 251 (1995), 267-388 MR1312334 (95m:81055)

[8] M.G. Darboux, “Sur une proposition relative aux equations lineaires”, Comptes Rendus Acad.Sci. Paris 94 (1882), 1456-1459

[9] B. Demircioglu, S. Kuru, M. Onder and A. Vercin, “Two families of superintegrable andisospectral potentials in two dimensions”, J. Math. Phys. 43 (2002), 2133-2150 MR1893664(2003c:81066)

[10] C. Gu, H. Hu and Z. Zhou, “Darboux transformations in integrable systems”, (MathematicalPhysics Studies 26, Springer, Dordrecht, The Netherlands, 2005) MR2174988 (2006i:37141)

[11] M.V. Ioffe, D.N. Nishnianidze and P.A. Valinevich, “New exactly solvable two-dimensionalquantum model not amenable to separation of variables”, J. Phys. A 43 (2010), 485303MR2738144 (2011k:81118)

[12] M.V. Ioffe, J. Negro, L.M. Nieto and D.N. Nishnianidze, “New two-dimensional integrablequantum models from SUSY intertwining”, J. Phys. A 39 (2006), 9297-9308 MR2247508(2007e:81028)

[13] G. Junker “Supersymmetric methods in quantum and statistical physics”, (Springer, Berlin,1995) MR1415615 (99a:81048)

[14] S. Kuru, A. Tegmen and A. Vercin, “Intertwined isospectral potentials in an arbitrary di-mension”, J. Math. Phys. 42 (2001), 3344-3360 MR1845193 (2002j:81091)

[15] V.B. Matveev and M.A. Salle, “Darboux transformations and solitons”, (Springer, Berlin,1991) MR1146435 (93d:35136)

[16] A. Schulze-Halberg, “Higher-order Darboux transformations in two dimensions, linearizableauxiliary equations and invariant potentials”, J. Math. Phys. 52 (2011), 083505 (9pp)

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Department of Mathematics and Actuarial Science, Indiana University Northwest,

3400 Broadway, Gary, Indiana 46408



Elliptic beta integrals and solvable models of statisticalmechanics

V. P. Spiridonov

Abstract. The univariate elliptic beta integral was discovered by the authorin 2000. Recently Bazhanov and Sergeev have interpreted it as a star-trianglerelation (STR). This important observation is discussed in more detail in con-nection to author’s previous work on the elliptic modular double and supersym-metric dualities. We describe also a new Faddeev-Volkov type solution of STR,connections with the star-star relation, and higher-dimensional analogues ofsuch relations. In this picture, Seiberg dualities are described by symmetriesof the elliptic hypergeometric integrals (interpreted as superconformal indices)which, in turn, represent STR and Kramers-Wannier type duality transfor-mations for elementary partition functions in solvable models of statisticalmechanics.

Contents

1. The simplest elliptic hypergeometric integrals2. The elliptic beta integral STR solution and star-star relation3. A hyperbolic beta integral STR solution4. Partition functions5. ConclusionAcknowledgmentsAppendix A. The modified q-gamma functionAppendix B. General multiple gamma functionsReferences

1. The simplest elliptic hypergeometric integrals

In the present paper we discuss relations between a new class of special func-tions, called elliptic hypergeometric functions, and solvable models of statisticalmechanics. We describe the most complicated known integrable systems definedon 2d (two-dimensional) lattices representing continuous spin generalizations of thewell known Ising model and its various extensions. Actually, these novel integrablemodels correspond to some discretized 2d quantum field theories. Also we indicate

2010 Mathematics Subject Classification. Primary 82B23, Secondary 33E99.Key words and phrases. Elliptic beta integrals, integrable systems, statistical mechanics.Work was supported in part by Russian foundation for basic research (RFBR grant no.

09-01-00271).


181

182 V. P. SPIRIDONOV

connections with the 4d supersymmetric field theories, where elliptic hypergeomet-ric integrals have found recently the major application. We start from a brieftechnical introduction to the needed results on special functions and discuss thephysical systems they apply to in the following chapters.

General theory of elliptic hypergeometric integrals was formulated in [S1, S3,S5]. We skip the structural definition of these integrals and refer for the corre-sponding details to a reasonably short survey given in [S9].

Let us denote

(z; q)∞ =

∞∏k=0

(1− zqk), |q| < 1, z ∈ C,

the standard infinite q-product and

Γ(z; p, q) =∞∏

i,j=0

1− z−1pi+1qj+1

1− zpiqj, |p|, |q| < 1, z ∈ C∗,

the standard elliptic gamma function. Below we use the conventions

Γ(a, b; p, q) := Γ(a; p, q)Γ(b; p, q), Γ(az±1; p, q) := Γ(az; p, q)Γ(az−1; p, q),

Γ(az±1y±1; p, q) := Γ(azy; p, q)Γ(az−1y; p, q)Γ(azy−1; p, q)Γ(az−1y−1; p, q).

One has the symmetry Γ(z; p, q) = Γ(z; q, p) and the inversion formula

Γ(a,

pq

a; p, q

)= 1, or Γ(a, a−1; p, q) =

1

θ(a; p)θ(a−1; q),

which follows from the difference equations

Γ(qz; p, q) = θ(z; p)Γ(z; p, q), Γ(pz; p, q) = θ(z; q)Γ(z; p, q),

where

θ(z; p) = (z; p)∞(pz−1; p)∞

is a theta function. The standard odd Jacobi theta function has the form [WW]

θ1(u|τ ) = −θ11(u) = −∑k∈Z

eπiτ(k+1/2)2e2πi(k+1/2)(u+1/2)

= ip1/8e−πiu(p; p)∞θ(e2πiu; p),

where we denoted p = e2πiτ .The univariate elliptic beta integral [S1] forms a cornerstone of a new power-

ful class of exactly computable integrals. It is described by the following explicitformula

(1.1) κ

∫T

∏6j=1 Γ(tjz

±1; p, q)

Γ(z±2; p, q)

dz

iz=

∏1≤j<k≤6

Γ(tjtk; p, q),

where T is the unit circle with positive orientation,

κ =(p; p)∞(q; q)∞

4π,

and six complex parameters tj , j = 1, . . . , 6, satisfy the inequalities |tj | < 1 andthe balancing condition

(1.2)6∏

j=1

tj = pq.

ELLIPTIC BETA INTEGRALS AND SOLVABLE MODELS 183

We use the word “integral” in two meanings. When referred to the exactlycomputable cases, like (1.1) or the standard Euler beta integral lying on its bottom,it means either the function defined by the left-hand side or, more often, the wholeidentity. In other cases it means an integral representation for a function of interestor a class of functions with common structure.

As shown in [S3], the left-hand side of relation (1.1) serves as the orthogonalitymeasure for the most general known family of biorthogonal functions with theproperties characteristic to classical orthogonal polynomials (Chebyshev, Hermite,Laguerre, Jacobi, . . . , Askey-Wilson polynomials). In the same paper the ellipticbeta integral has been generalized to the following function

(1.3) V (t1, . . . , t8; p, q) = κ

∫T

∏8j=1 Γ(tjz

±1; p, q)

Γ(z±2; p, q)

dz

iz,

where |tj | < 1 and∏8

j=1 tj = (pq)2. This is a natural elliptic analogue of the Gausshypergeometric function since its features generalize most of the special functionproperties of the 2F1-series [S5, S9]. For tjtk = pq, j �= k, V -function reducesto the elliptic beta integral and, for this reason, it can be called the elliptic betaintegral of a higher order.

In [S4], the author has introduced the following universal integral transforma-tion for functions analytical in the vicinity of the unit circle T:

(1.4) g(w; t) = κ

∫T

Δ(t;w, z; p, q)f(z; t)dz

iz,

where the kernel

(1.5) Δ(t;w, z; p, q) := Δ(t;w, z) = Γ(tw±1z±1; p, q), |t| < 1,

is a particular product of four elliptic gamma functions. In [SW], it was shown thatthis integral transformation obeys the key property making it very similar to theFourier transformation. Namely, its inverse is obtained essentially by the reflectiont→ t−1.

An explicit example of the pair of functions g(w; t) and f(z; t) in (1.4) canbe easily found from the elliptic beta integral. Indeed, let us denote t5 = tw andt6 = tw−1 (so that t2

∏4j=1 tj = pq). Then,

f(z; t) =

∏4j=1 Γ(tjz

±1; p, q)

Γ(z±2; p, q),(1.6)

g(w; t) = Γ(t2; p, q)∏

1≤i<j≤4

Γ(titj ; p, q)

4∏j=1

Γ(ttjw±1; p, q),(1.7)

where |tw±1|, |tj | < 1.Because of the permutational symmetry, any of the original variables tj can be

associated with the distinguished parameter t. After fixing t1 = sy, t2 = sy−1 andt3 = rx, t4 = rx−1, one can rewrite the elliptic beta integral in the form∫

T

ϕ(z)Δ(r;x, z)Δ(s; y, z)Δ(t;w, z)dz

iz

= χ(r, s, t)Δ(rs;x, y)Δ(rt;x,w)Δ(st; y, w),(1.8)


where rst = ±√pq and

ϕ(z) =(p; p)∞(q; q)∞4πΓ(z±2; p, q)

=1

4π(p; p)∞(q; q)∞θ(z2; p)θ(z−2; q),

χ(r, s, t) = Γ(r2, s2, t2; p, q).(1.9)

A key application of definition (1.4) consists in the construction of a tree ofidentities for multiple elliptic hypergeometric integrals with many parameters [S4].Using one of the corresponding symmetry transformations, the following relationhas been derived in [S8]

(1.10) φ(x; c, d|ξ; s) = κ

∫T

R(c, d, a, b;x,w|s)φ(w; a, b|ξ; s)dwiw

,

where the “basis vector” φ has the form

(1.11) φ(w; a, b|ξ; s) = Γ(saξ±1, sbξ±1,

√pq

abw±1ξ±1; p, q),

and the “rotation” integral operator kernel is

R(c, d, a, b;x,w|s) = 1

Γ( pqab ,abpq , w

±2; p, q)

× V

(sc, sd,

√pq

cdx,

√pq

cdx−1,

pq

as,pq

bs,

√ab

pqw,

√ab

pqw−1; p, q

).

The function φ is a generalization of the kernel Δ(t;x, z), since for ab = pq/s2 onehas the reduction

φ(w; a,pq

as2|ξ; s) = Δ(s;w, ξ).

Using the Δ-kernel, relation (1.10) was rewritten also in [S8] in a more compactform

Δ(α;x, ξ)Δ(β; y, ξ) = κ

∫T

r(α, β, γ, δ;x, y; t, w)Δ(γ; t, ξ)Δ(δ;w, ξ)dw

iw,

r(α, β, γ, δ;x, y; t, w) =1

Γ(δ±2, w±2; p, q)V

(αx±1, βy±1,

pq

γt±1,

w±1

δ

),

where αβ = γδ and

V

(αx±1, βy±1,

pq

γt±1,

w±1

δ

)= κ

∫T

Δ(α;x, z)Δ(β; y, z)Δ( pqγ ; t, z)Δ( 1δ ;w, z)

Γ(z±2; p, q)

dz

iz.

Here we use the condensed notation for parameters of the V -function: V (. . . αx±1

. . .) = V (. . . αx, αx−1 . . .).The function φ emerges also in the context of the Sklyanin algebra [Sk] (the

algebra of the Yang-Baxter equation solutions),

SαSβ − SβSα = i(S0Sγ + SγS0),

S0Sα − SαS0 = iJβ − Jα

Jγ(SβSγ + SγSβ),(1.12)

where Jα are the structure constants and (α, β, γ) is any cyclic permutation of(1, 2, 3). Namely, one has to consider the generalized eigenvalue problems Aφ =λBφ, where A and B are linear combinations of four generators Sa, a = 0, 1, 2, 3,


and λ is a spectral parameter. The function φ is defined uniquely up to multipli-cation by a constant with the help of two such equations using a pair of Sklyaninalgebras forming an elliptic modular double [S8]. This algebra represents an ellipticextension of the Faddeev modular double [F], but there are actually two differentmodular doubles at the elliptic level which obey different sets of involutions.

Relevance of the Sklyanin algebra in this setting was noticed first by Rains[R1]. For special quantized values of the parameters, the φ-function reduces tothe intertwining vectors of Takebe [T], which were used by Rosengren in [Ros]for the derivation of a discrete spin version of relation (1.10). In our case both

Casimir operators of the algebra (1.12), K0 =∑3

a=0 S2a and K2 =

∑3α=1 JαS

2α, take

continuous values, i.e. we deal with the continuous spin representations related tothe integral operator form of the Yang-Baxter equation.

The following scalar product has been introduced in [S8]

(1.13) 〈χ, ψ〉 = κ

∫T

χ(z)ψ(z)

Γ(z±2; p, q)

dz

iz.

It has been shown that both the V -function itself and the φ-vectors form biorthog-onal systems of functions with respect to this measure. In particular, one has therelation

κ

∫T

φ(eiϕ′; pq

c ,pqd |ξ; s−1)φ(eiϕ; c, d|ξ; s)Γ(ξ±2; p, q)

dξ

iξ

=2π

(p; p)∞(q; q)∞Γ

(pq

cd,cd

pq, e±2iϕ; p, q

)√1− v2 δ(v − v′),(1.14)

where v = cosϕ, v′ = cosϕ′, and δ(v) is the Dirac delta-function. (There is amissprint in formula (3.2) of [S8] which misses the first factor standing on theright-hand side of (1.14).) Positivity of the biorthogonality measure and of theφ-function corresponds to the unitarity of representations of the elliptic modulardouble. Setting cd = pq/s2, we obtain

(2κ)2∫ 1

−1

Δ(s−1; eiϕ′, eiχ)Δ(s; eiϕ, eiχ)

Γ(e±2iχ; p, q)

dX√1−X2

= Γ(s2, s−2, e±2iϕ; p, q

)√1− v2 δ(v − v′),(1.15)

where X = cosχ.The tetrahedral symmetry transformation for V -function, discovered in [S3],

can be rewritten in the following form:

V (αx±1, βy±1, γw±1, δz±1) = Γ(α2, β2, γ2, δ2; p, q)Δ(αβ;x, y)Δ(γδ;w, z)

× V (√pqβ−1x±1,

√pqα−1y±1,

√pqδ−1w±1,

√pqγ−1z±1)(1.16)

= Δ(αγ;x,w)Δ(αδ;x, z)Δ(βγ; y, w)Δ(βδ; y, z)

× V (βx±1, αy±1, δw±1, γz±1)(1.17)

= Γ(α2, β2, γ2, δ2; p, q)Δ(αβ;x, y)Δ(αγ;x,w)Δ(αδ;x, z)Δ(βγ; y, w)(1.18)

×Δ(βδ; y, z)Δ(γδ;w, z)V (√pqα−1x±1,

√pqβ−1y±1,

√pqγ−1w±1,

√pqδ−1z±1),

where αβγδ = ±pq. The latter two transformations are obtained by repeated ap-plication of the first relation in combination with permutation of the parameters.The full symmetry group of the V -function is the Weyl group W (E7) for the ex-ceptional root system E7 [R3]. Therefore, there are 72 = dimW (E7)/S8 relations


similar to (1.16), (1.17), (1.18), we just picked up three of them by breaking theS8 permutational symmetry and gathering the elliptic gamma functions into theΔ-blocks.

The outstanding physical application of the elliptic beta integral has been dis-covered by Dolan and Osborn [DO]. They have shown that the simplest supercon-formal (topological) indices of N = 1 supersymmetric field theories coincide withknown elliptic hypergeometric integrals. Exact computability or the Weyl groupsymmetry transformations of such integrals describe the Seiberg duality of N = 1theories [Sb], since they prove coincidence of the corresponding superconformalindices.

In this picture, the left-hand side of the univariate elliptic beta integral eval-uation formula (1.1) describes the superconformal index of the supersymmetricquantum chromodynamics with SU(2) gauge group and SU(6) flavor group. Thistheory has one vector superfield (gauge fields) in the adjoint representation of SU(2)and a set of chiral superfields (matter fields) in the fundamental representation ofSU(2) × SU(6). The elementary particles representing these fields describe thespectrum of the theory in the high energy limit, where the coupling constant isvanishing due to the asymptotic freedom. In the deep infrared region the theory isstrongly coupled, all colored particles confine, and one has the Wess-Zumino typemodel for mesonic fields lying in the 15-dimensional totally antisymmetric tensorrepresentation of SU(6). The superconformal index of the latter theory is de-scribed by the right-hand side expression of formula (1.1). This construction givesa group-theoretical interpretation of the elliptic beta integral. After renormalizingthe parameters tk = (pq)1/6yk, the balancing condition takes the form

∏6k=1 yk = 1,

which is nothing else than the unitarity condition for the maximal torus variablesof the group SU(6). This is the simplest example of the Seiberg duality discoveredin [Sb]. Further detailed investigation of such interrelations and their consequencescan be found in [SV1], where many new elliptic beta integrals on root systems havebeen conjectured and many new supersymmetric dualities have been found.

The elliptic hypergeometric integrals emerge also in the context of the relativis-tic Calogero-Sutherland type models [S7]. However, the first non-trivial exampleof the elliptic hypergeometric functions was found from the exactly solvable mod-els of statistical mechanics. Namely, in [FT] Frenkel and Turaev have shown thatthe Boltzmann weights (elliptic 6j-symbols) of the RSOS models of Date et al[DJKMO], generalizing Baxter’s eight-vertex model [Bax1], are determined byparticular values of the terminating 12V11 elliptic hypergeometric series (in mod-ern notations of [S9]). The same series has been found by Zhedanov and theauthor [SZ] in a completely different setting, as a particular solution of the Laxpair equations for a classical discrete integrable system. In [S2, S3], a familyof meromorphic functions obeying a novel two-index biorthogonality relation hasbeen discovered. It was explicitly conjectured in [S2] that these functions deter-mine a new family of solutions of the Yang-Baxter equation for discrete spin models.Since the V (t1, . . . , t8; p, q) function is an integral generalization of the latter func-tions, in [S6] it was conjectured that the V -function determines a solution of theYang-Baxter equation. A simple connection of the terminating 12V11-series and V -function with the Yang-Baxter equation for RSOS models was discussed in [KS].Recently, Bazhanov and Sergeev [BS] have shown that the elliptic beta integral canbe rewritten as a star-triangle relation (STR) which yields a new two-dimensional


solvable model of statistical mechanics. This is a new important application ofintegral (1.1) which is described in the next section. In this paper we show thatthe symmetry transformations for the V -function have similar interpretation as thestar-star relations. Moreover, we conjecture that all known exact formulas for el-liptic hypergeometric integrals describing the Seiberg duality transformations (atthe level of superconformal indices) [SV1], in turn, represent STR and Kramers-Wannier type duality transformations [KW, W] for elementary partition functionsin solvable models of statistical mechanics [Bax2].

2. The elliptic beta integral STR solution and star-star relation

In [BS], Bazhanov and Sergeev have interpreted the elliptic beta integral eval-uation formula as a star-triangle relation which gave a new solution of this relation.In order to describe it, let us introduce the parameter η related to the bases p andq as

e−2η = pq

and pass to the additive notation

z = eiu, x→ eix, y → eiy, w → eiw.

Introduce also the exponential form of the parameters

r = e−α, s = eα+γ−η, t = e−γ ,

so that the balancing condition r2s2t2 = pq is satisfied automatically. Finally,denote

(2.1) W (α;x, u) := Δ(eα−η; eix, eiu).

Then relation (1.8) can be rewritten as∫ 2π

0

S(u; p, q)W (η − α;x, u)W (α+ γ; y, u)W (η − γ;w, u)du

= χ(α, γ; p, q)W (α; y, w)W (η − α− γ;x,w)W (γ;x, y),(2.2)

where

S(u; p, q) =(p; p)∞(q; q)∞

4πθ(e2iu; p)θ(e−2iu; q),(2.3)

χ(α, γ; p, q) = Γ(r2, s2, t2; p, q).(2.4)

As observed in [BS], equality (2.2) is nothing else than the star-triangle relationplaying an important role for solvable models of statistical mechanics. It is sym-bolized by figure 1 given below, where the black vertex of the star-shaped figure onthe left-hand side means the integration over u-variable with the weight S(u), andW -weights are associated with the edges connecting the black vertex with whiteones. On the right-hand side one has the product of three W -weights connectingonly white vertices.

Suppose we have a two dimensional square lattice with spin variables a, b, c, . . .sitting at vertices. One associates the self-interaction energy S(a) with each spin(vertex). For each horizontal bond connecting spins a and b the energy contributionis given by the Boltzmann weight Wfg(a, b), and the energy contribution from each

vertical bond connecting spins b and d is given by the weight W fg(b, d). Thevariables f and g are called rapidities. Then, as described in detail by Baxter in


Figure 1. The star-triangle relation.

[Bax3, Bax4], the general STR for these quantities have the following functionalequations form:∑

d

S(d)W fg(d, b)Wfh(c, d)W gh(a, d) = RfghWfg(c, a)W fh(a, b)Wgh(c, b),∑d

S(d)W fg(b, d)Wfh(d, c)W gh(d, a) = RfghWfg(a, c)W fh(b, a)Wgh(b, c).(2.5)

The second equation is satisfied automatically if the Boltzmann weights are sym-metric in spin variables

Wfg(a, b) = Wfg(b, a), W fg(a, b) = W fg(b, a).

Usually the normalization constants factorize, Rfgh = rghrfg/rfh. Then the weightssatisfy the unitarity relation of the form∑

d

S(d)W fg(a, d)W gf (d, b) =rfgrgfS(a)

δab

and the reflection equation Wfg(a, b)Wgf (a, b) = 1.A subclass of solutions of (2.5) emerges from the weights depending only on

differences of the rapidities,

(2.6) Wfg(a, b) = W (f − g; a, b), W fg(a, b) = W (η − f + g; a, b),

where the parameter η is called the crossing parameter. Then the precise identifica-tion of equality (2.2) with (2.5) is reached after setting α = f−g, γ = g−h (so thatf − h = α+ γ), equating S(d) to S(u; p, q) and Rfgh to χ(α, γ; p, q) functions, andfixing appropriately the range of summation (integration) over the variable d = u.We call (2.1), (2.3), (2.4) the elliptic beta integral STR solution. As shown in [BS],it generalizes many known solvable models of statistical mechanics [Bax2]: the Isingmodel, Ashkin-Teller, chiral Potts, Fateev-Zamolodchikov ZN -model, Kashiwara-Miwa and Faddeev-Volkov models. Moreover, as will be shown below, it comprisesalso a new Faddeev-Volkov type integrable system with continuous spins.

There is direct relation between spin systems on lattices of three types — thehoneycomb, triangular, and rectangular lattices. Indeed, one can start from thehoneycomb lattice, as depicted on the left-hand side of figure 2. Applying the star-triangle transformation to each black vertex one transforms the whole honeycomblattice to the triangular one [W]. In a similar way, one can apply STR to each whitevertex and obtain another triangular lattice having only black vertices. This is quiteevident and does not require further explanations. However, further transformationof the triangular lattice to the square one is more tricky.

Consider the left-hand side of figure 3. Take the horizontal line in the middleof the drawn piece of the lattice. Pick up the triangles above and below it which


Figure 2. A honeycomb-triangluar lattice transformation in-duced by the star-triangle relation.

intersect only at one point lying on this line (they are shown in bold lines). Apply tothem the triangle-star relation replacing them by stars and continue this procedureup and down line-by-line of the resulting lattice. As a result, one obtains eventuallythe square lattice. Taking into account the nontrivial χ-multiplier in STR, onecan thus connect partition functions of the square lattice model to the partitionfunctions of two other types of models.

Figure 3. A triangluar-rectangular lattice transformation in-duced by the star-triangle relation.

In [BS], the parameters x and u in (2.1) were considered as true spin variables.However, because of the x→ −x and u→ −u symmetries, the Boltzmann weightsW and S depend on their trigonometric combinations. Therefore one can count asthe true spin variables U = cosu, X = cosx, etc, with their values ranging from -1to 1. The change of the variables in the measure is elementary∫

T

f(12(z + z−1)

)dziz

=

∫ 2π

0

f(cosu)du = 2

∫ 1

−1

f(U)dU√1− U2

.

The Boltzmann weight W (α;x, u) satisfies the reflection symmetry

W (α;x, u)W (−α;x, u) = 1,


following from the reflection equation for the elliptic gamma function. In terms ofthe spin variables X = cosx and Y = cos y the unitarity relation takes the form∫ 1

−1

S(u; p, q)W (η − α;x, u)W (η + α; y, u)dU√1− U2

=Γ(e2α, e−2α; p, q)

S(x; p, q)

√1−X2 δ(X − Y ).(2.7)

This equality has been established by the author in [S8]. Note that positivity ofthe Boltzmann weights S(u; p, q) and W (α;x, u) corresponds to the unitarity of theelliptic modular double representations [S8]. In particular, they are positive forx, u ∈ [0, 2π], real α such that |√pqeα| < 1, and

1) p∗ = p, q∗ = q, or 2) p∗ = q.

At the level of superconformal indices, relations similar to (2.7) describe the Seibergdualities for gauge field theories with equal number of colors and flavors and thechiral symmetry breaking [SV2].

Relation (2.2) is not changed if one replaces W and χ by

W (α;x, u) =W (α;x, u)

m(α),

χ(α, γ; p, q) =m(α)m(γ)m(η − α− γ)

m(η − α)m(η − γ)m(α+ γ)χ(α, γ; p, q)(2.8)

for arbitrary normalizing factor m(α).The star-triangle relation is one of the three known forms of the Yang-Baxter

equation. The second, probably the most popular form, is the vertex type relationsymbolically written in terms of the R-matrices as

(2.9) R(12)(λ)R(13)(λ+ μ)R(23)(μ) = R(23)(μ)R(13)(λ+ μ)R(12)(λ),

where λ and μ are spectral parameters. The third type is referred to as the IRF(interaction around the face) Yang-Baxter equation. The star-star relation, whichwas discussed in detail in [Bax3], belongs to the latter type of equations and hasthe form ∑

g

S(g)W1(a, g)W2(b, g)W3(c, g)W4(d, g)

= Rm(b, c)p(a, b)

m(a, d)p(c, d)

∑g

S(g)W ′1(a, g)W

′2(b, g)W

′3(c, g)W

′4(d, g),(2.10)

where Wj(a, b),W′j(a, b),m(b, c), p(a, b) are two-spin Boltzmann weights and S(g)

is the spin self-interaction weight (it was omitted in formula (1.1) of [Bax3]). Theleft-hand side can be interpreted as an elementary partition function for a systemof four spins a, b, c, d sitting in four square vertices connected by edges to the sping sitting in the square center, and the summation is going over the values of thecentral spin, see figure 4 below. The right hand side has a similar interpretation ofa statistical sum multiplied by the additional Boltzmann weights associated withopposite edges of the square (a, b, c, d). Formula (2.10) can be thought of as ageneralized Kramers-Wannier duality transformation [KW, W].

Relation (2.10) should be compared with the V -function symmetry transfor-mations written in the form (1.16), (1.17), and (1.18). Some of them coincide with


a b

d c

W

W

W

W

g

a b

d c

W’

W’

W’

W’

g

p

m

Figure 4. A star-star relation for the square lattice. AdditionalBoltzmann weights p and m are indicated by edges connectingcorresponding vertices on the right-hand side.

(2.10) after appropriate identifications of the Boltzmann weights. For instance,equation (1.16) corresponds to the choice

W1(a, g) = Δ(α;x, g), W ′1(a, g) = Δ(

√pqβ−1;x, g),

W2(b, g) = Δ(β; y, g), W ′2(b, g) = Δ(

√pqα−1; y, g),

W3(c, g) = Δ(√pqγ;w, g), W ′

3(c, g) = Δ(δ−1;w, g),

W4(d, g) = Δ(√pqδ; z, g), W ′

4(d, g) = Δ(γ−1; z, g),

where αβγδ = 1, g is the integration variable for the V -function, and S(g) =κ/Γ(g±2; p, q). Other factors have the form

R =Γ(α2, β2; p, q)

Γ(γ−2, δ−2; p, q), m(b, c) = m(a, d) = 1,

p(a, b) = Δ(αβ;x, y), p(c, d) = Δ(αβ;w, z).(2.11)

A similar interpretation is valid for relation (1.17). It corresponds to the choice

W1(a, g) = Δ(α;x, g), W ′1(a, g) = Δ(β;x, g),

W2(b, g) = Δ(√pqγ;w, g), W ′

2(b, g) = Δ(√pqδ;w, g),

W3(c, g) = Δ(β; y, g), W ′3(c, g) = Δ(α; y, g),

W4(d, g) = Δ(√pqδ; z, g), W ′

4(d, g) = Δ(√pqγ; z, g),

where, again, αβγδ = 1 and S(g) = κ/Γ(g±2; p, q). As to other factors, R = 1 and

m(b, c) = Δ(√pqβγ; y, w), m(a, d) = Δ(

√pqβγ;x, z),

p(a, b) = Δ(√pqαγ;x,w), p(c, d) = Δ(

√pqαγ; y, z).(2.12)

There are three star-star relations for the Ising type models listed in [Bax3] asequations (2.16), (5.1), and (5.2). Our first option (2.11) corresponds to relation(5.2) in [Bax3]. Relations (2.16) and (5.2) in [Bax3] are obtained from each otherby a reflection with respect to the lattice square diagonal (b, d). Our second option(2.12) corresponds to relation (5.1) in [Bax3] with nonconstant p- and m-weights.However, we have the third nontrivial form of the symmetry transformation forthe V -function (1.18). It corresponds to a more complicated type of the star-star


relation∑g

S(g)W1(a, g)W2(b, g)W3(c, g)W4(d, g)(2.13)

= Rm(b, c)p(a, b)t(a, c)

m(a, d)p(c, d)t(b, d)

∑g

S(g)W ′1(a, g)W

′2(b, g)W

′3(c, g)W

′4(d, g),

where t(a, c) is a new diagonal Boltzmann weight. Explicitly, we have

W1(a, g) = Δ(α;x, g), W ′1(a, g) = Δ(

√pqα−1;x, g),

W2(b, g) = Δ(β; y, g), W ′2(b, g) = Δ(

√pqβ−1; y, g),

W3(c, g) = Δ(√pqγ;w, g), W ′

3(c, g) = Δ(γ−1;w, g),

W4(d, g) = Δ(√pqδ; z, g), W ′

4(d, g) = Δ(δ−1; z, g),

where αβγδ = 1. Other factors in (2.13) are R = Γ(α2, β2; p, q)/Γ(γ−2, δ−2; p, q)and

m(b, c) = Δ(√pqβγ; y, w), m(a, d) = Δ(

√pqβγ;x, z),

p(a, b) = Δ(αβ;x, y), p(c, d) = Δ(αβ;w, z),

t(a, c) = Δ(√pqαγ;x,w), t(b, d) = Δ(

√pqαγ; y, z).

Perhaps, this type of the star-star relation was not considered in the literaturebefore. Note that all such relations represent symmetry groups of the partitionfunctions. In the case of V -function this is W (E7), i.e. one has much biggersymmetry than that seen explicitly in the chosen spin system interpretation. Wehave described thus a new (elliptic hypergeometric) class of solutions of the star-starrelation which should lead to new solvable models of statistical mechanics similarto the checkerboard Ising model. Known systems of such type were investigatedin detail in [BSt]. A natural general conclusion from our consideration is that thesymmetry of STR can be richer than a direct sum of symmetries of the Boltzmannweights and the lattice.

3. A hyperbolic beta integral STR solution

We describe now another solution of the star-triangle relation associated withthe modified form of the elliptic beta integral when one of the bases p or q can lieon the unit circle [DS2]. It simplifies also consideration of the degeneration limitsto q-beta integrals of the Mellin-Barnes type (hyperbolic beta integrals).

First we describe the modified elliptic gamma function introduced in [S3]. It isconvenient to use additive notation and introduce three pairwise incommensuratequasiperiods ω1, ω2, ω3 together with the definitions

q = e2πiω1ω2 , p = e2πi

ω3ω2 , r = e2πi

ω3ω1 ,

q = e−2πiω2ω1 , p = e−2πi

ω2ω3 , r = e−2πi

ω1ω3 .(3.1)

Here q, p, and r are particular (τ → −1/τ ) modular transformations of q, p, andr. Assume that Im(ω1/ω2), Im(ω3/ω1), Im(ω3/ω2) > 0, or |q|, |p|, |r| < 1. Then themodified elliptic gamma function is constructed as a product of two elliptic gammafunctions

G(u;ω1, ω2, ω3) = Γ(e2πiuω2 ; p, q)Γ(re−2πi u

ω1 ; r, q)

= e−πi3 B3,3(u;ω)Γ(e−2πi u

ω3 ; r, p),(3.2)


where B3,3(u;ω) is the third diagonal Bernoulli polynomial (for the general defini-tion of such polynomials, see Appendix A),

B3,3

(u+

3∑n=1

ωn

2;ω

)=

u(u2 − 14

∑3k=1 ω

2k)

ω1ω2ω3.

The G(u;ω)-function satisfies the following system of three linear difference equa-tions of the first order

G(u+ ω1;ω) = θ(e2πiuω2 ; p)G(u;ω),

G(u+ ω2;ω) = θ(e2πiuω1 ; r)G(u;ω),

G(u+ ω3;ω) = e−πiB2,2(u;ω)G(u;ω),

where B2,2(u;ω) is the second diagonal Bernoulli polynomial,

B2,2(u;ω) =u2

ω1ω2− u

ω1− u

ω2+

ω1

6ω2+

ω2

6ω1+

1

2.

The second equality in (3.2) follows from the fact that both expressions for G(u;ω)

satisfy the above set of equations and the normalization G( 12∑3

k=1 ωk;ω) = 1.It is easy to see that G(u;ω) is well defined for |p|, |r| < 1 and |q| ≤ 1, the

|q| = 1 case being permitted in difference from the Γ(z; p, q)-function. Evidently,we have the symmetry relation

G(u;ω1, ω2, ω3) = G(u;ω2, ω1, ω3)

and the reflection equation

G(a;ω)G(b;ω) = 1, a+ b =3∑

k=1

ωk.

For Im(ω1/ω2) > 0, we can take the limit ω3 →∞ in such a way that

Im(ω3/ω1), Im(ω3/ω2)→ +∞and p, r → 0. Then,

(3.3) limp,r→0

G(u;ω) = γ(u;ω1, ω2) =(e2πiu/ω1 q; q)∞(e2πiu/ω2 ; q)∞

.

For Re(ω1),Re(ω2) > 0 and 0 < Re(u) < Re(ω1 + ω2) this γ-function has thefollowing integral representation

(3.4) γ(u;ω1, ω2) = exp

(−∫R+i0

eux

(1− eω1x)(1− eω2x)

dx

x

),

which shows that γ(u;ω1, ω2) is a meromorphic function of u even for ω1/ω2 > 0,when |q| = 1 and the infinite product representation (3.3) is not valid any more.The inversion relation for this function has the form

γ(u;ω1, ω2)γ(ω1 + ω2 − u;ω1, ω2) = eπiB2,2(u;ω).

For more details on this function see Appendix A.Let Im(ω1/ω2) ≥ 0 and Im(ω3/ω1), Im(ω3/ω2) > 0, and let six complex param-

eters gk, k = 1, . . . , 6, satisfy the constraints Im(gk/ω3) < 0 and

(3.5)6∑

k=1

gk = ω1 + ω2 + ω3.


Then [DS2],

(3.6)

∫ ω3/2

−ω3/2

∏6k=1 G(gk ± u;ω)

G(±2u;ω) du = κ∏

1≤k<l≤6

G(gk + gl;ω),

where the integration goes along the straight line segment connecting points −ω3/2and ω3/2, and

(3.7) κ =−2ω2(q; q)∞

(q; q)∞(p; p)∞(r; r)∞.

Here and below we use the shorthand notation

G(a± b;ω) := G(a+ b, a− b;ω) := G(a+ b;ω)G(a− b;ω).

The proof of equality (3.6) is rather simple. It is necessary to substitute in itthe second form of G(u;ω)-function (3.2), check that all exponential factors canceland, after a change of notation, the formula reduces to the standard elliptic betaintegral.

Let us introduce the crossing parameter

η = −1

2

3∑k=1

ωk

and denote

(3.8) g1,2 = −α± x, g3,4 = α+ γ − η ± y, g5,6 = −γ ± w,

so that the balancing condition (3.5) is satisfied automatically. Introduce also themodified Boltzmann weight, or the kernel for the modified form of the integraltransformation (1.4),

W ′(α;x, u) = G(α− η ± x± u;ω).

Then relation (3.6) can be rewritten as∫ ω3/2

−ω3/2

φ(u;ω)W ′(η − α;x, u)W ′(α+ γ; y, u)W ′(η − γ;w, u)du

= χ(α, γ;ω)W ′(α; y, w)W ′(η − α− γ;x,w)W ′(γ;x, y),(3.9)

where

φ(u;ω) =1

κG(±2u;ω) =1

κe−πiB2,2(2u;ω1,ω2)θ(e−4πiu/ω2 ; p)θ(e−4πiu/ω1 ; r),

χ(α, γ;ω) = G(−2α,−2γ, 2α+ 2γ − 2η;ω).(3.10)

Substituting the second form of the modified elliptic gamma function, we find

W ′(α;x, u) = exp

(−4πi

3

(B3,3(α− η;ω) +

3α(x2 + u2)

ω1ω2ω3

))×Δ

(e2πi(η−α);

x

ω3,u

ω3; p, r

).(3.11)

We see that this Boltzmann weight is obtained from (2.1) after a reparametrizationof variables and multiplication by an exponential of a quadratic polynomial of thespin variables. This means that there exists a nontrivial symmetry transformationof the star-triangle relation modifying its solutions in the described way.


The distinguished property of the modified elliptic beta integral is that it iswell defined for |q| = 1. Therefore the limit ω3 → ∞ leads to q-beta integrals welldefined in this regime as well. Let Re(ω1), Re(ω2) > 0. Then, for ω3 → +i∞,one has p, r → 0 and G(u;ω) goes to γ(u;ω1, ω2)-function. Let us substitute

g6 =∑3

k=1 ωk − A in formula (3.6), where A =∑5

k=1 gk, and apply the inversionformula to the corresponding modified elliptic gamma function. Then the formallimit ω3 → +i∞ reduces this integration formula to∫ +i∞

−i∞

∏5j=1 γ(gk ± u;ω)

γ(±2u,A± u;ω)du = −2ω2

(q; q)∞(q; q)∞

∏1≤j<k≤5 γ(gj + gk;ω)∏5

k=1 γ(A− gk;ω),(3.12)

where the integration contour is the straight line for Re(gk) > 0 or the Mellin-Barnestype contour, if these restrictions for parameters are violated. Let us remind alsothat

(q; q)∞(q; q)∞

=

√−iω1

ω2e

πi12

(ω1ω2

+ω2ω1

),

where√−i = e−πi/4 since for ω1/ω2 = ia, a > 0, the square root should be positive.

Let us introduce parameter g6 anew (it should not be confused with the previousvariable g6 which we have eliminated) using the condition

(3.13)

6∑k=1

gk = ω1 + ω2

(note the difference with (3.5)). Now we can apply the inversion formula to γ-functions to move some of them from the denominator of the integral kernel to itsnumerator. It is convenient here to define the hyperbolic gamma function γ(2)(u):

(3.14) γ(2)(u;ω) = e−πi2 B2,2(u;ω)γ(u;ω).

Then, after the change of the integration variable u = iz, the integral (3.12) takesthe compact form∫ ∞

−∞

∏6j=1 γ

(2)(gk ± iz;ω)

γ(2)(±2iz;ω) dz = 2√ω1ω2

∏1≤j<k≤6

γ(2)(gj + gk;ω).(3.15)

Validity of the described limit ω3 →∞ at the level of integrals was rigorouslyjustified in [R2] using a slightly different notation. Integral (3.15) was proven first(using a different approach) by Stokman [St] who called it the hyperbolic betaintegral. We followed the formal limiting procedure suggested in [DS2].

Similar to (3.8), let us fix the parameters as

(3.16) g1,2 = −α± ix, g3,4 = α+ γ − η ± iy, g5,6 = −γ ± iw

with the crossing parameter η = −(ω1+ω2)/2. Then formula (3.15) can be rewrittenas a star-triangle relation∫ ∞

−∞S(z)W (η − α;x, z)W (α+ γ; y, z)W (η − γ;w, z)dz

= χ(α, γ)W (α; y, w)W (η − α− γ;x,w)W (γ;x, y),(3.17)


where

W (α;x, z) = γ(2)(α− η ± ix± iz;ω),

S(z) =1

2√ω1ω2γ(2)(±2iz;ω) =

2 sinh 2πzω1

sinh 2πzω2√

ω1ω2,

χ(α, γ) = γ(2)(−2α,−2γ, 2α+ 2γ − 2η;ω).(3.18)

These Boltzmann weights are positive for real x, z, α, η < α < −η, η < 0, andeither real ω1,2 or ω∗

1 = ω2.Consider a particular reduction of integration formula (3.15). For this we re-

place parameters gj → gj + iμ, j = 1, 2, 3, and gj → gj − iμ, j = 4, 5, 6. Since theintegrand is symmetric in z we can rewrite the left-hand side as

2

∫ ∞

0

∏3j=1 γ

(2)(gj + iμ± iz, gj+3 − iμ± iz;ω)

γ(2)(±2iz;ω) dz

= 2

∫ ∞

−μ

3∏j=1

γ(2)(gj − iz, gj+3 + iz;ω)ρ1(z)ρ2(z)dz,

where

ρ1(z) =e−2π(z+μ)(ω−1

1 +ω−12 )

γ(2)(±2i(z + μ);ω)→

μ→+∞1

and

ρ2(z) = e2π(z+μ)(ω−11 +ω−1

2 )3∏

j=1

γ(2)(gj + 2iμ+ iz, gj+3 − 2iμ− iz;ω)

→μ→+∞

eπ

ω1ω2

(−2μ(ω1+ω2)+

i2

∑3j=1

(g2j+3−g2

j+(gj−gj+3)(ω1+ω2)

))(1 + o(1)).

On the right-hand side we find

2√ω1ω2

3∏j=1

6∏k=4

γ(2)(gj + gk;ω)ρ3(g),

where

ρ3(g) =∏

1≤j<k≤3

γ(2)(gj + gk + 2iμ, gj+3 + gk+3 − 2iμ;ω)

→μ→+∞

eπi2

∑1≤j<k≤3

(B2,2(gj+3+gk+3−2iμ)−B2,2(gj+gk+2iμ)

)(1 + o(1)).

One can check that the leading asymptotics of ρ3(g) coincides with that of theρ2(z)-function. Taking the limit μ → +∞, which is uniform, one comes to thefollowing exact integration formula [B]

(3.19)

∫ ∞

−∞

3∏j=1

γ(2)(gj − iz, gj+3 + iz;ω)dz =√ω1ω2

3∏j=1

6∏k=4

γ(2)(gj + gk;ω),

where∑6

k=1 gk = ω1 + ω2.


Let us change notation for the integral parameters

g1 = −α+ ix, g2 = α+ γ − η + iy, g3 = −γ + iw,

g4 = −α− ix, g5 = α+ γ − η − iy, g6 = −γ − iw,

where η = −(ω1+ω2)/2 is a crossing parameter; the balancing condition∑6

k=1 gk =−2η is satisfied then automatically. Now we can rewrite equality (3.19) as the star-triangle relation∫ ∞

−∞W (η − α;x, z)W (α+ γ; y, z)W (η − γ;w, z)

dz√ω1ω2

= χW (α; y, w)W (η − α− γ;x,w)W (γ;x, y),(3.20)

where the Boltzmann weight is defined as

W (α;x, z) = γ(2)(α− η ± i(x− z);ω)

and the normalization constant is

(3.21) χ = γ(2)(−2α,−2γ, 2α+ 2γ − 2η;ω).

Note that W (α;x, y) = W (α; y, x) and W (α;x, y) > 0 in the same domain ofparameters as before. Denoting ω = b, ω2 = b−1, and α = −(b + b−1)θ/(2π) onecan see that W (α;x, z) coincides with the Boltzmann weight of the Faddeev-Volkovmodel [FV, VF] denoted as Wθ(x − z) in [BMS] (our η differs by sign from thedefinition chosen in [BMS]) up to some normalization factor Fθ.

We thus see that the Faddeev-Volkov model solution of the star-triangle relation[VF] is a particular case of our hyperbolic beta integral STR solution (3.18). Thefact that the left-hand side of STR for the Faddeev-Volkov model represents aparticular limiting case of the elliptic beta integral was known to the author alreadyin 2008. After seeing [BMS] and understanding this fact, the author was interestedwhether a similar interpretation exists for the elliptic beta integral itself. However,this idea was not developed further, partially because the origin of the normalizingfactor Fθ given in [BMS] was not understood at that time. Fortunately, Bazhanovand Sergeev have independently answered this question in [BS].

4. Partition functions

The partition function of a homogeneous two dimensional discrete spin systemon the square lattice with the Boltzmann weights W (α;ui, uj) (2.1) and S(uj) (2.3)has the form

Z =

∫ ∏(ij)

W (α;ui, uj)∏(kl)

W (η − α;uk, ul)∏m

S(um)dum,

where the first product is taken over the horizontal edges (ij), the second productgoes over all vertical edges (k, l), and the third product (in m) is taken over allinternal vertices of the lattice. Let us take the elliptic beta integral STR solutionof [BS] and consider the contribution to Z coming from a particular vertex usurrounded by the vertices u1, u2, u3, u4:∫ 2π

0

S(u)W (α;u1, u)W (α;u, u3)W (η − α;u2, u)W (η − α;u, u4)du.


Substituting explicit expressions for the weights, one can easily see that this integralis equal to the elliptic hypergeometric function V (t1, . . . , t8; p, q) described above(1.3) with the following restricted set of parameters

{t1, t2, t3, t4} = {eα−ηe2πiu1 , eα−ηe−2πiu1 , eα−ηe2πiu3 , eα−ηe−2πiu3},{t5, t6, t7, t8} = {e−αe2πiu2 , e−αe−2πiu2 , e−αe2πiu4 , e−αe−2πiu4}.

In total, there are 5 independent parameters, instead of 7 for generic V -function(in addition to the bases p and q). Therefore we conclude that the full partitionfunction Z is given by an elliptic hypergeometric integral constructed as a towerof intertwined (restricted) elliptic analogues of the Gauss hypergeometric functionsimilar to the Bailey tree for integrals [S4].

According to the general reflection method used in [BS], the leading asymp-totics of the partition function for two-dimensional N×M lattice when its size goesto infinity, N,M →∞, has the form

Z =N,M→∞

m(α)NM ,

where m(α) is the normalizing factor for Boltzmann weights which guarantees thaton the right-hand side of STR the χ-multiplier (2.8) is equal to unity, χ = 1. Thiscondition is satisfied if

(4.1)m(α)

m(η − α)Γ(e−2α; p, q) = 1, or m(α+ η) = Γ(e2α; p, q)m(−α).

Let us introduce the function(4.2)

M(x; p, q, t) = exp( ∑

n∈Z/{0}

(√pqtx)n

n(1− pn)(1− qn)(1 + tn)

)=

Γ(xt√pqt; p, q, t2)

Γ(x√pqt; p, q, t2)

,

where

Γ(z; p, q, t) =∞∏

j,k,l=0

(1− ztjpkql)(1− z−1tj+1pk+1ql+1), |p|, |q|, |t| < 1,

is the second order elliptic gamma function satisfying the t-difference equation

Γ(tz; p, q, t) = Γ(z; p, q)Γ(z; p, q, t).

The reflection equation Γ(z−1; p, q, t) = Γ(pqtz; p, q, t) leads to the equality

M(x−1; p, q, t)M(x; p, q, t) = 1.

It is easy also to check validity of the functional equation

M(x; p, q, t)M(t−1x; p, q, t) = Γ(x

√pq

t; p, q

),

which is equivalent to (4.1) after fixing t = pq and x = e2α. Therefore we find theneeded normalizing function

(4.3) m(α) = M(e2α; p, q, pq), m(α)m(−α) = 1.

The function − logm(α) defines thus the free energy per edge of the integrablelattice model under consideration.


Now we discuss the partition function for the general hyperbolic beta integralsolution of the star-triangle relation (3.18). The needed normalization constantm(α) is found from the equation

(4.4)m(α)

m(η − α)γ(2)(−2α;ω) = 1, or m(α/2 + η) = γ(2)(α;ω)m(−α/2),

where η = −(ω1 + ω2)/2.Let us define the function

μ(u;ω1, ω2, ω3) =γ(3)(u+ 1

2

∑3k=1 ωk + ω3;ω1, ω2, 2ω3)

γ(3)(u+ 12

∑3k=1 ωk;ω1, ω2, 2ω3)

,

where γ(3)-function is the hyperbolic gamma function of the third order defined inAppendix B. Using the integral representation for it, we can write

(4.5) μ(u;ω) = exp(− πia

6−∫R+i0

evx

(eω1x − 1)(eω2x − 1)(eω3x + 1)

dx

x

),

where v = u+∑3

k=1 ωk/2 and

a = B3,3(v + ω3;ω1, ω2, 2ω3)−B3,3(v;ω1, ω2, 2ω3)

=3

2ω1ω2

(u2 − ω2

1 + ω22 + 3ω2

3

12

)For a special choice of the third quasiperiod variable ω3 = ω1 + ω2, this functionappeared for the first time in [LZ].

Using the reflection equation

γ(3)(

3∑k=1

ωk − u;ω1, ω2, ω3) = γ(3)(u;ω1, ω2, ω3)

and the difference equation

γ(3)(u+ ω3;ω1, ω2, ω3) = γ(2)(u;ω1, ω2)γ(3)(u;ω1, ω2, ω3),

one can easily check that μ(u;ω)μ(−u;ω) = 1 and

μ(u;ω)μ(u− ω3;ω) = γ(2)(u+1

2(ω1 + ω2 − ω3);ω1, ω2).

The latter relation coincides with equation (4.4) for u = 2α and ω3 = ω1 + ω2.Therefore we find the free energy per edge as − logm(α), where

(4.6) m(α) = μ(2α;ω1, ω2, ω1 + ω2).

By construction this function satisfies also the reflection equation m(α)m(−α) = 1.Denoting ω1 = b, ω2 = b−1 and substituting the infinite product representation ofthe γ(3)-function given in Appendix B, we find the expression

m(α) = exp

(−πiα2 − πi

24(1− 2(b+ b−1)2)

)× (qe2πiu/b; q2)∞

(qe2πiub; q2)∞

∞∏j,k=0

1 + eπiu/(b+b−1)pj+1q2k

1− eπiu/(b+b−1)pj+1q2k,(4.7)

where it is assumed that |q| < 1, q = e2πib2

, q = e−2πi/b2 , and p = e−πi/(1+b2).We turn now to the Faddeev-Volkov model solution of STR (3.20). In this case

we have no self-interaction of the spins sitting in lattice vertices, and the Boltzmann


weights attached to edges are simplified. But the partition function asymptoticsis the same as in the previous case, since evidently the normalizing constant m(α)is found from the same equation (4.4). The free energy per edge for this modelwas computed already by Bazhanov, Mangazeev, and Sergeev in [BMS], where theBoltzmann weights normalizing factor was denoted as Fθ. Comparing this constantwith our m(α), we see that they coincide for α = −(b + b−1)θ/(2π), as necessary.However, our infinite product representation ofm(α) in (4.7) differs drastically fromthat given in [BMS] (which was the source of author’s old time confusion).

5. Conclusion

After the discovery of elliptic hypergeometric integrals, for a long time the au-thor was drawing attention of experts (including the second author of [BS]) in two-dimensional conformal field theory and solvable models of statistical mechanics fora potential emergence of such functions in these fields. The connection between theelliptic beta integral and the star-triangle relation found in [BS] and the star-starrelation described above confirms this expectation. However, the nature appearedto be much richer than it was imagined in [S2, S6, S9]. As mentioned already, theDolan-Osborn discovery of a stunningly unexpected coincidence of elliptic hyper-geometric integrals with superconformal (topological) indices in four dimensionalsupersymmetric gauge theories strongly pushed forward the development of thetheory and raised many interesting open questions [DO, SV1]. The interpretationof exact computability of the elliptic beta integrals as the confinement phenome-non in quantum field theory is a new type of conceptual perception of the exactmathematical formulas.

As to the models considered in this paper, we have described a generalizationof the Faddeev-Volkov solution of STR [VF] with the continuous spin variablestaking values on the real line, which was not considered in [BS]. It has somenontrivial self-interaction energy for each vertex and a more complicated form ofthe Boltzmann weights for edges, though the free energy per edge appears to bethe same as in the Faddeev-Volkov model. In [VF] the Yang-Baxter equation wasproved using the quantum pentagonal relation [FKV]. It would be interesting tointerpret in a similar way the model we have described here. Some time ago theauthor has tried to find an elliptic analogue of the pentagon relation in analogywith the constructions described in [V], but could not do it yet. Clearly the ellipticbeta integral gives already an analytic form of that wanted operator relation, butit is hard to formulate it in terms of the commutation relations of some explicitoperators.

In [FV], Faddeev and Volkov have considered a lattice Virasoro algebra anddescribed an integrable model in the discrete 2d space-time (it was discussed alsoin detail in [FKV]). The elliptic beta integral yields more general solutions ofSTR than that of [VF], and it is natural to ask for explicit realization of thecorresponding models similar to [FV]. During the work on [SV1], G. Vartanovand the author have suggested that there should exist some elliptic deformation ofthe primary fields Vα(z) built from free 2d bosonic fields (in the spirit similar to thesituation discussed in [SWy]) such that the three point correlation function wouldbe given by the elliptic beta integral and the four point function would be describedby the V -function satisfying the elliptic hypergeometric equation [S9] (so that the


tetrahedral symmetries of the V -function would describe the s-t channels duality).Unfortunately, such hypothetical results are not conceivable at the present moment.

From the point of view of superconformal indices the partition function as-sociated with the elliptic beta integral solution of STR looks as a superconfor-mal index for a particular SU(2)-quiver gauge theory on a two dimensional lat-tice. Recently there was a great deal of activity on interrelations between four-dimensional super-Yang-Mills theories and two-dimensional field theories, see, e.g.,[AGT, CNV, GPRR, NS, SWy]. In this framework, the elliptic hypergeomet-ric integrals describing superconformal indices of N = 2 quiver gauge theories havebeen interpreted by Gadde et al in [GPRR] as correlation functions of some 2dtopological quantum field theories.

Therefore it is natural to expect that superconformal indices of all four di-mensionanl CFTs are related to discretizations of 2d CFT models and other inte-grable systems. A connection of the Yang-Baxter moves with the Seiberg dualityhas been briefly discussed in [HV]. In this context, superconformal indices of allquiver gauge theories should correspond to full partition functions of some spin sys-tems. In view of the abundance of supersymmetric dualities and rich structure ofthe corresponding superconformal indices (twisted partition functions) [SV1], theauthor considers the present moment only as a beginning of uncovering new two-dimensional and higher-dimensional integrable models hidden behind the elliptichypergeometric functions.

For instance, the elliptic Selberg integral defined on the BCn root system reads[DS1, S9]:

κn

∫Tn

∏1≤j<k≤n

Γ(tz±1j z±1

k ; p, q)

Γ(z±1j z±1

k ; p, q)

n∏j=1

∏6m=1 Γ(tmz±1

j ; p, q)

Γ(z±2j ; p, q)

dzjizj

=

n∏j=1

⎛⎝Γ(tj ; p, q)

Γ(t; p, q)

∏1≤m<s≤6

Γ(tj−1tmts; p, q)

⎞⎠ ,(5.1)

where |t|, |tm| < 1, t2n−2∏6

m=1 tm = pq, and

κn =(p; p)n∞(q; q)n∞

(4π)nn!.

After some work, this formula can be given the STR type shape∫[0,2π]n

S(u; t, p, q)W (η − α;x,u)W (α+ γ; y,u)W (η − γ;w,u)[du],

= Wt(α; y, w)Wt(η − α− γ;w, x)Wt(γ;x, y),(5.2)

where we denoted

[du] = κn

n∏j=1

Γ(t; p, q)duj

Γ(tj ; p, q),

and the crossing parameter η is defined as

e−2η = pqtn−1.

The Boltzmann weights have the form

S(u; t, p, q) =∏

1≤j<k≤n

Γ(te±iuj±iuk ; p, q)

Γ(e±iuj±iuk ; p, q)

n∏j=1

1

Γ(e±2iuj ; p, q)(5.3)


and

W (α;x,u) :=1

m(α)

n∏j=1

Γ(√pqeαe±ixe±iuj ; p, q),(5.4)

Wt(α;x, y) :=1

m(α)

n∏j=1

Γ(√pqeαtj−

n+12 e±ixe±iy; p, q),(5.5)

and satisfy the reflection relations

W (α;x,u)W (−α;x,u) = 1, Wt(α;x, y)Wt(−α;x, y) = 1.

The normalization constant m(α) for n > 1 has a substantially more complicatedform than that for n = 1. To describe it we introduce the function

M(x; p, q, t, s) =Γ(xts2, xt1−ns; p, q, t, s2)

Γ(xts, xt1−ns2; p, q, t, s2),

a particular ratio of four elliptic gamma functions of the third order. More precisely,one has

Γ(z; p, q, t, s) :=

∞∏i,j,k,l=0

1− z−1pi+1qj+1tk+1sl+1

1− zpiqjtksl

for z ∈ C∗, |p|, |q|, |t|, |s| < 1, with the reflection equation Γ(z, pqtsz−1; p, q, t, s) = 1and the difference equation Γ(sz; p, q, t, s) = Γ(z; p, q, t)Γ(z; p, q, t, s). Then,

(5.6) m(α) = M(e2α; p, q, t, pqtn−1),

with the standard reflection relation m(α)m(−α) = 1.Let us discuss a physical meaning of the obtained model. Consider a honeycomb

lattice on the plane with two types of vertices – black and white with two adjacentvertices always being of different color, see the left-hand side of figure 2. Intoeach white vertex we put an independent single component continuous spin x.Into each black vertex we put n independent spins uj , j = 1, . . . , n, or one n-dimensional spin with n continuous components. These “spins” are quite differentfrom those of the Ising model where they take only the values +1 and −1 (i.e., theyrepresent the fields and not the compact spins). In different words, one associatesto each black vertex the SP (2n)-group space related to the root system BCn. Weassociate with each black vertex the self-interaction Boltzmann weight (5.3) withan additional interaction between “spin” components in the internal space. To eachbond connecting “black-and-white” vertices we attach the Boltzmann weight (5.4).Then, on the left-hand side of (5.2) we have the partition function of an elementarycell with the black vertex in the center and the integral taken over the uj-spinvalues. If we apply this star-triangle relation to each black vertex we come to adifferent spin system associated with the plain triangular lattice having only thewhite vertices with the bond Boltzmann weights described by the function (5.5),see the right-hand side of figure 2. Such a transformation of lattices looks quitesimilar to a transformation of the honeycomb-triangular Ising systems considered in[W]. Perhaps there exists also another STR type duality transformation involvingonly the white vertices (with some self-interaction) allowing for a transition to yetanother triangular lattice system. In addition to this uncertainty, it remains alsounclear the free energy per edge of which model is described by the function (5.6).

Positivity of the Boltzmann weights of this model can be analyzed along thelines of elliptic modular double involutions discussed in [S8]. In particular, these


weights are clearly positive for x, uj ∈ [0, 2π], real t, α and ρ := |√pqeα| < 1,

ρ < |t|n−12 < ρ−1, with either p∗ = p, q∗ = q or p∗ = q. For n > 1 the crossing

parameter η looks like an arbitrary free variable, not related to other parametersof the system, but, in fact, it is essentially equivalent to the coupling constant t foruj-spins. If t = 1, relation (5.2) reduces to n-th power of the standard STR.

Define the BCn-root system generalization of the V -function:

I(t1, . . . , t8; t; p, q) =∏

1≤j<k≤8

Γ(tjtk; p, q, t)(5.7)

× κn

∫Tn

∏1≤j<k≤n

Γ(tz±1j z±1

k ; p, q)

Γ(z±1j z±1

k ; p, q)

n∏j=1

∏8k=1 Γ(tkz

±1j ; p, q)

Γ(z±2j ; p, q)

dzjizj

,

where parameters t, t1, . . . , t8 ∈ C, satisfy |t|, |tj | < 1, and t2n−2∏8

j=1 tj = p2q2

constraints. As shown by Rains [R3], this function obeys the same W (E7) Weylgroup of symmetries as in the n = 1 case. The key transformation has the form

(5.8) I(t1, . . . , t8; t; p, q) = I(s1, . . . , s8; t; p, q),

where{sj = ρ−1tj , j = 1, 2, 3, 4sj = ρtj , j = 5, 6, 7, 8

; ρ =

√t1t2t3t4pqt1−n

=

√pqt1−n

t5t6t7t8, |t|, |tj |, |sj | < 1.

Introduce variables xj by relation t(n−1)/4tj = (pq)1/4e2πixj , so that the balancing

condition becomes∑8

j=1 xj = 0. Then (5.8) describes the invariance of the integralI with respect to the Weyl reflection

x→ Sv(x) = x− 2〈x, v〉〈v, v〉 v, x, v ∈ R8,

where 〈x, v〉 =∑8

k=1 xkvk is the scalar product and the vector v has componentsvk = 1/2, k = 1, 2, 3, 4, and vk = −1/2, k = 5, 6, 7, 8. Together with the groupS8 permuting the parameters xj , this transformation generates full exceptionalreflection group W (E7).

Equality (5.8) can be rewritten in the star-star relation form∫[0,2π]n

S(u; t, p, q)W (η − α;x,u)W (η − β; y,u)W (γ;w,u)W (δ; z,u)[du]

= RPt(α+ β;x, y)

Pt(α+ β;w, z)

∫[0,2π]n

S(u; t, p, q)(5.9)

×W (β;x,u)W (α; y,u)W (η − δ;w,u)W (η − γ; z,u)[du],

where α+ β = γ + δ and

R =

n−1∏l=0

Γ(t−le−2α, t−le−2β ; p, q)

Γ(t−le−2γ , t−le−2δ; p, q), Pt(α;x, y) =

n−1∏l=0

Γ(t−le−αe±ix±iy; p, q).

In complete analogy with n = 1 case (1.16), (1.17), (1.18), one can obtain two otherdifferently looking star-star relations for n > 1 by an iterative application of thisformula after permutations of parameters.

Using the matrix integral representations for elliptic hypergeometric integrals,in [SV1] relation (5.8) was shown to describe a new electric-magnetic dualitybetween two four-dimensional N = 1 supersymmetric Yang-Mills theories with


the gauge group G = SP (2n). Namely, the electric theory has the flavor groupSU(8)× U(1); it contains the vector superfield in the adjoint representation of G,one chiral scalar multiplet in the fundamental representations of G and SU(8), andthe field described by the antisymmetric tensor of the second rank of G. The mag-netic theory has the flavor group SU(4)l × SU(4)r × U(1)B × U(1) and a similarset of quantum fields, as well as 2n additional gauge invariant mesonic fields —the antisymmetric tensors of SU(4)-flavor subgroups. The elliptic Selberg integral(5.1) describes the confinement phenomenon in the SP (2n) super-Yang-Mills gaugetheory with 6 chiral superfields in the fundamental and 1 chiral superfield in theantisymmetric representations of SP (2n), respectively, — its dual magnetic phasecontains only a peculiar set of mesonic fields without local gauge symmetry.

From the present paper point of view relations (5.2) and (5.9) should have anappropriate physical interpretation in the context of discrete integrable models forn > 1 similar to the n = 1 case. We described already one possible honeycomblattice model that can be associated with the elliptic Selberg integral. The systemlying behind relation (5.9) resembles the checkerboard Ising model with the contin-uous spins. As an elementary cell one has a square with four white vertices (withthe single component spins x, y, . . . sitting in them) and one black vertex in thecenter (with the n-component spin u sitting in it and the integral taken over itsvalues), see figure 4. There are again three differently looking star-star relations forn > 1 obtained by repeated application of the same formula (5.9) in conjugationwith permutation of parameters, quite similar to the n = 1 case. Equality (5.2)can be considered then as their reduction to STR.

We did not discuss in this paper an important physical question of the existenceof phase transitions in the described models and the spectrum of scaling exponents.For clarifying this point it is necessary to single out the temperature like variableassociated with one of the parameters p or q [BS] and investigate the behavior of thepartition functions per edge (defined by m(α)’s) when the temperature varies fromlarge to small values. Since many known systems with nontrivial phase transitionsare represented by the limiting cases of the elliptic beta integral STR solutions, thereare nontrivial critical phenomena. However, their classification requires separateanalysis and lies beyond the scope of the present work.

In [SV1] a large number of proven and conjectural evaluation formulas forelliptic beta integrals on root systems and their nontrivial symmetry transformationanalogues for higher order integrals has been listed. Actually, it was conjecturedthat there exist infinitely many such integrals, and for each of them one can expectsuitable application in the context of solvable models of statistical mechanics andother types of integrable systems.

Acknowledgments

The results of this paper were partially reported at the Jairo Charris seminar(August 3–6, 2010, Santa Marta, Colombia). The hospitality of P. Acosta-Humanezduring this workshop is gratefully appreciated. The author is deeply indebtedto A.N. Kirillov, V.B. Priezzhev, I.P. Rochev, and G.S. Vartanov for stimulatingdiscussions. A.M. Povolotsky is thanked for teaching me the graphics drawing.Both referees are thanked for helping in improving the paper.


Appendix A. The modified q-gamma function

The function γ(u;ω1, ω2) (3.3) or its various transformed versions are referredto in different papers as the double sine function [KLS, PT], the non-compactquantum dilogarithm [F, FKV, PT, V, BS], the hyperbolic gamma function[Ru, B], or the modified q-gamma function [S9].

The functional equations satisfied by γ(u;ω) have the form

(A.1)γ(u+ ω1;ω)

γ(u;ω)= 1− e2πi

uω2 ,

γ(u+ ω2;ω)

γ(u;ω)= 1− e2πi

uω1 .

Using a modular transformation for theta functions one can derive another repre-sentation for γ(u;ω1, ω2) complementary to (3.3):

(A.2) γ(u;ω1, ω2) = eπiB2,2(u;ω) (e−2πiu/ω2q; q)∞

(e−2πiu/ω1 ; q)∞.

The non-compact quantum dilogarithm [F] in the notation of [BMS] (in [FKV]it was denoted as eb(z)) has the form

ϕ(z) = exp

(1

4

∫R+i0

e−2izw

sinh(wb) sinh(w/b)

)dw

w

= exp

(∫R+i0

ewu

(ewb − 1)(ew/b − 1)

)dw

w,

where

u =1

2(b+ b−1)− iz.

For q = e2πib2

, q = e−2πi/b2 and Im(b2) > 0, one can write

ϕ(z) =(e2πibu; q)∞(e2πiu/bq; q)∞

=(−q1/2e2πbz ; q)∞(−q1/2e2πz/b; q)∞

.

Therefore,

ϕ(z) = γ(12(b+ b−1)− iz; b, b−1

)−1.

The Sb(u) function used in [PT] coincides with

γ(2)(u;ω1, ω2) = e−πi2 B2,2(u;ω1,ω2)γ(u;ω1, ω2)

for ω1 = b and ω2 = b−1, and another similar function of [PT] is

Gb(u) = e−πi12 (3+b2+b−2)γ(u; b, b−1).

The γ(z)-function used in [V] coincides with the infinite products ratio on theright-hand side of (A.2) for z = ω2u and τ = ω1/ω2.

For Re(ω1),Re(ω2) > 0 and 0 < Re(u) < Re(ω1+ω2) the function γ(2)(u;ω1, ω2)has the following integral representation

(A.3) γ(2)(u;ω1, ω2) = exp

(−PV

∫R

eux

(1− eω1x)(1− eω2x)

dx

x

),

where the principal value of the integral means PV∫R= 2−1(

∫R+i0

+∫R−i0

). Using

the fact that PV∫Rdx/xk = 0 for k > 1, one can write

γ(2)(u;ω1, ω2) = exp

(−∫ ∞

0

( sinh(2u− ω1 − ω2)x

2 sinh(ω1x) sinh(ω2x)− 2u− ω1 − ω2

2ω1ω2x

)) dx

x.


Comparing this expression with the hyperbolic gamma function Gh(z;ω) definedin [Ru], one can see that

Gh(z;ω) = γ(2)(12(ω1 + ω2)− iz;ω

), Re(ω1),Re(ω2) > 0.

Changing in (A.3) the sign of the integration variable x→ −x and simultaneouslyωk → −ωk, u→ −u, we find that

(A.4) γ(2)(u;ω1, ω2) = exp

(+PV

∫R

eux

(1− eω1x)(1− eω2x)

dx

x

),

where Re(ω1),Re(ω2) < 0 and Re(ω1 + ω2) < Re(u) < 0.The double sine function is defined as S2(u;ω) = 1/γ(2)(u;ω) and its properties

were described in detail in the Appendix of [KLS]. For the γ(2)-function we have

γ(2)(u;ω1, ω2)∗ = γ(2)(u∗;ω∗

1 , ω∗2), γ(2)(

ω1 + ω2

2± u;ω1, ω2) = 1,

and γ(2)(au; aω1, aω2) = γ(2)(u;ω1, ω2) for arbitrary complex a �= 0. After such arescaling in (A.4) with a = 2πi one gets the definition of the hyperbolic gammafunction given in [R2].

The asymptotics we are interested in for Im(ω1/ω2) > 0 have the form

limu→∞

eπi2 B2,2(u;ω)γ(2)(u;ω) = 1, for argω1 < arg u < argω2 + π,

limu→∞

e−πi2 B2,2(u;ω)γ(2)(u;ω) = 1, for argω1 − π < arg u < argω2.

Appendix B. General multiple gamma functions

Barnes introduced a multiple zeta function as the following m-fold series [Bar]

ζm(s, u;ω) =∞∑

n1,...,nm=0

1

(u+Ω)s, Ω = n1ω1 + . . .+ nmωm,

where s, u ∈ C. This series converges for Re(s) > m under the condition that all ωj

lie in one half-plane defined by a line passing through zero. Because of the latterrequirement, the sequences n1ω1+ . . .+nmωm do not have accumulation points onthe finite plane for any nj → +∞. It is convenient to assume for definiteness thatRe(ωj) > 0.

The function ζm(s, u;ω) satisfies equations

(B.1) ζm(s, u+ ωj ;ω)− ζm(s, u;ω) = −ζm−1(s, u;ω(j)), j = 1, . . . ,m,

where ω(j) = (ω1, . . . , ωj−1, ωj+1, . . . , ωm) and ζ0(s, u;ω) = u−s. The Barnes mul-tiple gamma function is defined by the equality

Γm(u;ω) = exp(∂ζm(s, u;ω)/∂s)∣∣s=0

.

It satisfies finite difference equations

(B.2) Γm(u+ ωj ;ω) =1

Γm−1(u;ω(j))Γm(u;ω), j = 1, . . . ,m,

where Γ0(u;ω) := u−1.The multiple sine-function is defined as

Sm(u;ω) =Γm(

∑mk=1 ωk − u;ω)(−1)m

Γm(u;ω).


It is more convenient to work with the hyperbolic gamma function

γ(m)(u;ω) = Sm(u;ω)(−1)m−1

satisfying the equations

γ(m)(u+ ωj ;ω) = γ(m−1)(u;ω(j)) γ(m)(u;ω), j = 1, . . . ,m.

Note that the elliptic gamma function can be written as a special combination offour Barnes gamma functions of the third order [S9], and similar relations are validfor higher order elliptic gamma functions used in the present paper.

One can derive the integral representation [N]

γ(m)(u;ω) = exp

(−PV

∫R

eux∏mk=1(e

ωkx − 1)

dx

x

)= exp

(− πi

m!Bm,m(u;ω)−

∫R+i0

eux∏mk=1(e

ωkx − 1)

dx

x

)= exp

(πi

m!Bm,m(u;ω)−

∫R−i0

eux∏mk=1(e

ωkx − 1)

dx

x

),

where Re(ωk) > 0 and 0 < Re(u) < Re(∑m

k=1 ωk) and Bm,m are multiple Bernoullipolynomials defined by the generating function

(B.3)xmexu∏m

k=1(eωkx − 1)

=

∞∑n=0

Bm,n(u;ω1, . . . , ωm)xn

n!.

Infinite product representations for these functions have been derived in [N].In particular, for |p|, |q| < 1 and |r| > 1 we have

γ(3)(u;ω) = e−πi6 B3,3(u;ω)

∞∏j,k=0

(1− e2πiu/ω1 qj+1r−(k+1))(1− e2πiu/ω2pjqk)

1− e2πiu/ω3 pj+1r−k,

which is used in the main text after the reduction ω3 = 2(ω1 + ω2) (or p = q2,r = q−2, p = e−πiω2/(ω1+ω2)).

The functions m(α) (4.3), (4.6), and (5.6) defining the free energy per edge asdescribed in the main part of the paper are related to particular cases of the Lerchtype generalization of the Barnes zeta-function:

ζm(s, u;β;ω) =

∞∑n1,...,nm=0

∏mk=1 β

nk

k

(u+Ω)s, Ω = n1ω1 + . . .+ nmωm,

converging for all |βk| < 1, or Re(s) > m and |βk| = 1 (provided the same con-straints on ωj are valid as in the plain Barnes case). The univariate case, i.e. theproper Lerch zeta-function, is described, e.g., in [WW].

The function ζm(s, u;β;ω) satisfies the following set of finite difference equa-tions(B.4)βjζm(s, u+ ωj ;β;ω)− ζm(s, u;β;ω) = −ζm−1(s, u;β(j);ω(j)), j = 1, . . . ,m,

where ω(j) = (ω1, . . . , ωj−1, ωj+1, . . . , ωm), β(j) = (β1, . . . , βj−1, βj+1, . . . , βm),and ζ0(s, u;β;ω) = u−s.


Similar to the Barnes case, one can easily derive the integral representations

ζm(s, u;β;ω) =1

Γ(s)

∫ ∞

0

ts−1e−ut∏mk=1(1− βke−ωkt)

dt

=iΓ(1− s)

2π

∫CH

(−t)s−1e−ut∏mk=1(1− βke−ωkt)

dt,

where CH is the Hankel contour encircling the half-line [0,∞) counterclockwise,and using them analytically continue ζm-function in s and βk to different regionsof parameters. The βk-deformation of the Barnes multiple gamma function definedas Γm(u;β;ω) = exp(∂ζm(s, u;β;ω)/∂s)

∣∣s=0

satisfies the finite difference equations

(B.5) Γm(u+ ωj ;β;ω)βj =

1

Γm−1(u;β(j);ω(j))Γm(u;β;ω), j = 1, . . . ,m,

where Γ0(u;β;ω) := u−1.When βk are primitive roots of unity, βnk

k = 1, nk = 1, 2, 3, . . . , it is possible torewrite ζm(s, u;β;ω) as linear combinations of the standard Barnes zeta functions.It follows from the simple identity

1

1− βkz=

∏l=0,2,...,nk−1(1− βl

kz)

1− znk.

This allows expressing the functions like (4.5) as linear combinations of the standardBarnes gamma functions, which was used in the construction of infinite productrepresentations of the functions m(α) (4.3), (4.6), and (5.6). In particular, function(4.5) is emerging from m = 3 case with the choice β1 = β2 = 1, β3 = −1.

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Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

Current address: Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow reg.141980, Russia


Selected Titles in This Series

563 Primitivo B. Acosta-Humanez, Federico Finkel, Niky Kamran, and Peter J.Olver, Editors, Algebraic Aspects of Darboux Transformations, Quantum IntegrableSystems and Supersymmetric Quantum Mechanics, 2012

562 P. Ara, K. A. Brown, T. H. Lenagan, E. S. Letzter, J. T. Stafford, and J. J.Zhang, Editors, New Trends in Noncommutative Algebra, 2012

561 Oscar Blasco, Jose A. Bonet, Jose M. Calabuig, and David Jornet, Editors,Topics in Complex Analysis and Operator Theory, 2012

560 Weiping Li, Loretta Bartolini, Jesse Johnson, Feng Luo, Robert Myers, and J.Hyam Rubinstein, Editors, Topology and Geometry in Dimension Three, 2011

559 Guillaume Bal, David Finch, Peter Kuchment, John Schotland, PlamenStefanov, Gunther Uhlmann, and Gunther Uhlmann, Editors, Tomography andInverse Transport Theory, 2011

558 Martin Grohe and Johann A. Makowsky, Editors, Model Theoretic Methods inFinite Combinatorics, 2011

557 Jeffrey Adams, Bong Lian, and Siddhartha Sahi, Editors, Representation Theoryand Mathematical Physics, 2011

556 Leonid Gurvits, Philippe Pebay, J. Maurice Rojas, and David Thompson,Editors, Randomization, Relaxation, and Complexity in Polynomial Equation Solving,

2011

555 Alberto Corso and Claudia Polini, Editors, Commutative Algebra and ItsConnections to Geometry, 2011

554 Mark Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, SimeonReich, David Shoikhet, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems IV, 2011

553 Mark Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, SimeonReich, David Shoikhet, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems IV, 2011

552 Robert Sims and Daniel Ueltschi, Editors, Entropy and the Quantum II, 2011

551 Jesus Araujo-Gomez, Bertin Diarra, and Alain Escassut, Editors, Advances inNon-Archimedean Analysis, 2011

550 Y. Barkatou, S. Berhanu, A. Meziani, R. Meziani, and N. Mir, Editors,Geometric Analysis of Several Complex Variables and Related Topics, 2011

549 David Blazquez-Sanz, Juan J. Morales-Ruiz, and Jesus Rodrıguez Lombardero,

Editors, Symmetries and Related Topics in Differential and Difference Equations, 2011

548 Habib Ammari, Josselin Garnier, Hyeonbae Kang, and Knut Sølna, Editors,Mathematical and Statistical Methods for Imaging, 2011

547 Krzysztof Jarosz, Editor, Function Spaces in Modern Analysis, 2011

546 Alain Connes, Alexander Gorokhovsky, Matthias Lesch, Markus Pflaum,and Bahram Rangipour, Editors, Noncommutative Geometry and Global Analysis,2011

545 Christian Houdre, Michel Ledoux, Emanuel Milman, and Mario Milman,Editors, Concentration, Functional Inequalities and Isoperimetry, 2011

544 Carina Boyallian, Esther Galina, and Linda Saal, Editors, New Developments inLie Theory and Its Applications, 2011

543 Robert S. Doran, Paul J. Sally, Jr., and Loren Spice, Editors, Harmonic Analysison Reductive, p-adic Groups, 2011

542 E. Loubeau and S. Montaldo, Editors, Harmonic Maps and Differential Geometry,2011

541 Abhijit Champanerkar, Oliver Dasbach, Efstratia Kalfagianni, Ilya Kofman,Walter Neumann, and Neal Stoltzfus, Editors, Interactions Between HyperbolicGeometry, Quantum Topology and Number Theory, 2011

540 Denis Bonheure, Mabel Cuesta, Enrique J. Lami Dozo, Peter Takac, Jean VanSchaftingen, and Michel Willem, Editors, Nonlinear Elliptic Partial DifferentialEquations, 2011

SELECTED TITLES IN THIS SERIES

539 Kurusch Ebrahimi-Fard, Matilde Marcolli, and Walter D. van Suijlekom,Editors, Combinatorics and Physics, 2011

538 Jose Ignacio Cogolludo-Agustın and Eriko Hironaka, Editors, Topology ofAlgebraic Varieties and Singularities, 2011

537 Cesar Polcino Milies, Editor, Groups, Algebras and Applications, 2011

536 Kazem Mahdavi, Deborah Koslover, and Leonard L. Brown, III, Editors, Cross

Disciplinary Advances in Quantum Computing, 2011

535 Maxim Braverman, Leonid Friedlander, Thomas Kappeler, Peter Kuchment,Peter Topalov, and Jonathan Weitsman, Editors, Spectral Theory and GeometricAnalysis, 2011

534 Pere Ara, Fernando Lledo, and Francesc Perera, Editors, Aspects of OperatorAlgebras and Applications, 2011

533 L. Babinkostova, A. E. Caicedo, S. Geschke, and M. Scheepers, Editors, SetTheory and Its Applications, 2011

532 Sergiy Kolyada, Yuri Manin, Martin Moller, Pieter Moree, and Thomas Ward,Editors, Dynamical Numbers: Interplay between Dynamical Systems and NumberTheory, 2010

531 Richard A. Brualdi, Samad Hedayat, Hadi Kharaghani, Gholamreza B.Khosrovshahi, and Shahriar Shahriari, Editors, Combinatorics and Graphs, 2010

530 Vitaly Bergelson, Andreas Blass, Mauro Di Nasso, and Renling Jin, Editors,Ultrafilters across Mathematics, 2010

529 Robert Sims and Daniel Ueltschi, Editors, Entropy and the Quantum, 2010

528 Alberto Farina and Enrico Valdinoci, Editors, Symmetry for Elliptic PDEs, 2010

527 Ricardo Castano-Bernard, Yan Soibelman, and Ilia Zharkov, Editors, MirrorSymmetry and Tropical Geometry, 2010

526 Helge Holden, Helge Holden, and Kenneth H. Karlsen, Editors, NonlinearPartial Differential Equations and Hyperbolic Wave Phenomena, 2010

525 Manuel D. Contreras and Santiago Dıaz-Madrigal, Editors, Five Lectures inComplex Analysis, 2010

524 Mark L. Lewis, Gabriel Navarro, Donald S. Passman, and Thomas R. Wolf,Editors, Character Theory of Finite Groups, 2010

523 Aiden A. Bruen and David L. Wehlau, Editors, Error-Correcting Codes, FiniteGeometries and Cryptography, 2010

522 Oscar Garcıa-Prada, Peter E. Newstead, Luis Alvarez-Consul, Indranil Biswas,Steven B. Bradlow, and Tomas L. Gomez, Editors, Vector Bundles and ComplexGeometry, 2010

521 David Kohel and Robert Rolland, Editors, Arithmetic, Geometry, Cryptographyand Coding Theory 2009, 2010

520 Manuel E. Lladser, Robert S. Maier, Marni Mishna, and Andrew Rechnitzer,Editors, Algorithmic Probability and Combinatorics, 2010

519 Yves Felix, Gregory Lupton, and Samuel B. Smith, Editors, Homotopy Theory ofFunction Spaces and Related Topics, 2010

518 Gary McGuire, Gary L. Mullen, Daniel Panario, and Igor E. Shparlinski,Editors, Finite Fields: Theory and Applications, 2010

517 Tewodros Amdeberhan, Luis A. Medina, and Victor H. Moll, Editors, Gems inExperimental Mathematics, 2010

516 Marlos A.G. Viana and Henry P. Wynn, Editors, Algebraic Methods in Statisticsand Probability II, 2010

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/.

This volume represents the 2010 Jairo Charris Seminar in Algebraic Aspects of DarbouxTransformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics,which was held at the Universidad Sergio Arboleda in Santa Marta, Colombia.

The papers cover the fields of Supersymmetric Quantum Mechanics and Quantum In-tegrable Systems, from an algebraic point of view. Some results presented in this volumecorrespond to the analysis of Darboux Transformations in higher order as well as someexceptional orthogonal polynomials.

The reader will find an interesting Galois approach to study finite gap potentials.

American Mathematical Societywww.ams.org

IMA on the Webhttp://ima.usergioarboleda.edu.co

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