Book of Abstracts

International Workshop on Symmetries,Special Functions and Superintegrability

A conference in honor of

Prof. Mariano A. del Olmo

on the occasion of his 60th birthday

July 10–11, 2014

Scientific Committee

Angel Ballesteros (Burgos)

Jose F. Carinena (Zaragoza)

Enrico Celeghini (Firenze)

Pilar G. Estevez (Salamanca)

David J. Fernandez (Mexico DF)

Javier Negro (Valladolid)

Miguel A. Rodrıguez (Madrid)

Mariano Santander (Valladolid)

Organizing Committee

Oscar Arratia

Juan A. Calzada

Fernando Gomez-Cubillo

Manuel Gadella

Jose M. Izquierdo

Sengul Kuru

Javier Negro

Luis M. Nieto

Mariano Santander

Foreword

The “International Workshop on Symmetries, Special functions and Superintegrability” (S-3)is organized to celebrate the 60th birthday of Professor Mariano del Olmo, an internation-ally recognized expert in many branches of mathematical physics. He has made remarkablecontributions in this field since the the beginning of the eighties: quantum groups, coherentstates, integrable and superintegrable systems, representations of Lie groups and Lie alge-bras, special functions and other topics that show his wide range of interest.

The workshop program is precisely centered around some of these topics in which Ma-riano has been working for many years. It is a pleasure to see that many of his present andformer collaborators, either inside Spain or working in other countries, are making an effortto come to Valladolid and share with him this unique occasion.

The Organizing Committee

Valladolid, July 10th, 2014.

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Contents

Symmetries from the Solution ManifoldVictor Aldaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Time asymmetric quantum mechanicsArno Bohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A generalized approach to integrability by quadraturesJose F. Carinena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Penning trap in a rotating magnetic field: coherent states approachDavid J. Fernandez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 + 1 spectral problems arising from the Manakov-Santini systemPilar G. Estevez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Exceptional orthogonal polynomialsDavid Gomez–Ullate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

On the planar Demkov wave functionsMiguel A. Gonzalez Leon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Natural Limits of Electroweak Model and Contraction of its Gauge GroupNikolai Gromov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolutionequations in 2 + 1 dimensions

Faruk Gungor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Invariant solutions of the supersymmetric CPN−1 sigma model

Veronique Hussin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6On the structure of Schwinger’s measurement algebra: groups, groupoids and 2-groupoids

Alberto Ibort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Vacuum

Carlos Lopez-Lacasta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Scalar field fluctuations distorted by two pairs of δ − δ′ interactions

Juan Mateos Guilarte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Supersymmetric partners of the truncated harmonic oscillator

Vicente S. Morales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Exotic supersymmetry of reflectionless systems, and solitons

Mikhail Plyushchay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Some new superintegrable hamiltonian systems, solvable by factorization method

Orlando Ragnisco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Invariant transformations in Euclidean hyperkahler structures

Miguel A. Rodrıguez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8The role of indencomposable representations in statistical physics

Yvan Saint–Aubin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Haantjes Manifolds and Integrable Systems

Piergiulio Tempesta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9The WKB approximation in deformation quantization

Jaromir Tosiek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Position and momentum in the monochromatic Maxwell fish-eye

Kurt Bernardo Wolf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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List of invited talks

Symmetries from the Solution Manifold

Victor Aldaya

Instituto de Astrofısica de Andalucıa (Spain)

In this talk, extended symmetries (non-point symmetries) of a physical system are used tocharacterize the corresponding solution manifold by means of Noether invariants. This is thestarting point to the correct quantization in non-linear cases, where the success of CanonicalQuantization is not guaranteed. The use of the Poincare-Cartan form permits finding both thesymplectic structure on the solution manifold, through the Hamilton-Jacobi transformation,and the required symmetries, realized as Hamiltonian vector fields, associated with functionson the solution manifold (thus constituting an inverse of the Noether Theorem) lifted back tothe evolution space through the inverse of this Hamilton-Jacobi mapping. We present simplenon-trivial examples where the symmetries are found in a perturbative way.

Time asymmetric quantum mechanics

Arno Bohm

The University of Texas at Austin (USA)

The meaning of time asymmetry in quantum physics is discussed. On the basis of amathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamicalequations, the Schrodinger equation for states or the Heisenberg equation for observablesare given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space)of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time sym-metric group evolution. Scattering theory suggests to distinguish mathematically betweenstates (defined by a preparation apparatus) and observables (defined by a registration appa-ratus (detector)). If one requires that scattering resonances of width Γ and exponentiallydecaying states of lifetime τ = ~/Γ should be the same physical entities (for which there issufficient evidence) one is led to a pair of RHS’s of Hardy functions and connected with it,to a semigroup time evolution t0 ≤ t < ∞, with the puzzling result that there is a quantummechanical beginning of time, just like the big bang time for the universe, when it was aquantum system. The decay of quasi-stable particles is used to illustrate this quantum me-chanical time asymmetry. From the analysis of these processes, we show that the propertiesof rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetryin quantum mechanics.

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A generalized approach to integrability by quadratures

Jose F. Carinena

Universidad de Zaragoza (Spain)

The classical result of Lie on integrability by quadratures will be reviewed and somegeneralizations will be proposed. After a short review of the classical Lie theorem it will beshown that if we are able to construct in an iterative way a nested sequence of subalgebras Liof vector fields including the dynamical vector field and after some steps the Lie subalgebrais Abelian, then the dynamics is integrable by k−1 quadratures. The theory will be illustratedwith examples and an extension of the theorem where the Lie algebras are replaced by somedistributions will also be presented.

Penning trap in a rotating magnetic field: coherent states approach

David J. Fernandez

CINVESTAV (Mexico)

The quantum problem of a non-relativistic charged particle subject to the electric andmagnetic fields created by an ideal Penning trap plus a magnetic field rotating around thesymmetry axis is addressed. The transition to the rotating frame is performed in order toeliminate the initial time-dependence in the Hamiltonian; then a coherent states approachfor the resulting time-independent problem is implemented. It is determined the parametersdomain for the particle to be in the trap regime. Several physical quantities for the systembeing in a coherent state are also evaluated.

1 + 1 spectral problems arising from the Manakov-Santini system

Pilar G. Estevez

Universidad de Salamanca (Spain)

This talk deals with the spectral problem of the Manakov Santini system. The point Liesymmetries of the Lax pair have been identified. Several similarity reductions arise fromthese symmetries. An important benefit of our procedure is that the study of the Lax pairinstead of the partial differential equations yields the reductions of the eigenfunctions andalso the spectral parameter. Therefore, we have obtained five interesting spectral problemsin 1 + 1 dimensions.

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Exceptional orthogonal polynomials

David Gomez–Ullate

Universidad Complutense (Spain)

Exceptional orthogonal polynomials are extensions of classical orthogonal polynomialsin the sense that they are also eigenfunctions of a Sturm-Liouville problem although thedegree sequence has a finite number of gaps (missing degrees). Equivalently, in mathematicalphysics they appear as eigenfunctions of rational extensions of exactly solvable potentials.It has been conjectured that all exceptional orthogonal polynomials can be obtained throughDarboux-Crum transformations of their classical counterparts, and this conjecture has beenproved in the Hermite case using the connection between trivial monodromy potentials andDarboux transformations. In this talk we will review some of these recent results and we willmention current open problems.

On the planar Demkov wave functions

Miguel A. Gonzalez Leon

Universidad de Salamanca (Spain)

The quantum spectral problem of diatomic molecular ions, in the Born-Oppenheimerapproximation, is separable using spheroidal coordinates obtained by rotating the two-dimen-sional elliptic coordinates about the focal axis. Separability allows to split the Schrodingerequation in a system of ODEs consisting in two different Generalized Spheroidal equationsfor the three dimensional case, and the Razavy equation plus the Whittaker Hill equation inthe two dimensional one.

Demkov has searched for eigen-wave functions of the Hamiltonian corresponding to theenergy levels of a hydrogenoid atom. This procedure lead, for certain values of the nuclearcharges, to the construction of “finite” wave functions, the so-called Demkov wave functions.We have already explained this behaviour in terms of the Quasi-Exact Solvability of theunderlying equation, the Confluent Heun equation. We present here a complete analysis ofthe analogous planar Demkov wave functions.

Natural Limits of Electroweak Model and Contraction of its Gauge Group

Nikolai Gromov

Syktyvkar University (Russian Federation)

Energy is the natural and most importantt parameter in particle physics. The ElectroweakModel is a gauge theory based on the group SU(2) × U(1), acting in the boson, lepton andquark sectors, which are describes by vectors of C2 with different physical interpretations in

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different sectors. The contracted group SU(2; j) and its fundamental representation spaceC2(j) are obtained by the consistent rescaling of SU(2) and C2

z′(j) =

(jz′1

z′2

)=

(α jβ

−jβ α

)(jz1

z2

)= u(j)z(j)

when j → o. Transformation rules of the boson felds as well as of the left lepton and quarkfields are as follows: W±µ → jW±µ , Zµ → Zµ, Aµ → Aµ, νl → jνl, el → el, ul → jul,dl → dl. The right lepton and quark fields are SU(2)-singlets, i.e. scalars, and therefore arenot transformed. The parameter j is connected with the energy s in center-of-mass systemj2(s) = g

√s/mW , where mW is W -boson mass and g is constant. So contraction j → 0

corresponds to zero energy limit of the Electroweak Model.

Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolutionequations in 2 + 1 dimensions

Faruk Gungor

Istambul Technical University (Turkey)

In this talk I will consider variable coefficient generalizations of two well-known physi-cal models: Kadomtsev-Petviashvili (KP) and Zabolotskaya-Khoklov (or dispersionless KP)equation and discuss how the existence of Kac-Moody-Virasoro algebras as their point sym-metry algebras, when combined with equivalence transformations, can serve as an additionaltest for integrability.

Invariant solutions of the supersymmetric CPN−1 sigma model

Veronique Hussin

Universite de Montreal (Canada)

Constant curvature surfaces are constructed from the finite action solutions of the super-symmetric CPN−1 sigma model. They are shown to be related to invariant solutions of themodel. In the case of holomorphic solutions, we get an unicity theorem. Some new resultsare obtained for the case of non-holomorphic solutions.

On the structure of Schwinger’s measurement algebra: groups, groupoids and 2-groupoids

Alberto Ibort

Universidad Carlos III de Madrid (Spain)

The structure of Schwinger’s measurement algebra will be reviewed. We would pointout that it carries the structure of a 2-groupoid and the basic notions of groups, gropoids and2-groupoids will be discussed as well as some elementary examples.

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Vacuum

Carlos Lopez-Lacasta

Universidad de Alcala de Henares (Spain)

Vacuum as a Lorentz invariant fluid (distribution of density of particles in momentumspace), equivalently a relativistic ether, is the subject of this talk. In QM, the vacuum of vir-tual particles will be considered in a theory of hidden variables, not as a scientific proposal(because of the uncertainty principle), but as a (metaphysical) analogy between some QMphenomena and classical Brownian motion. In GR, there is wide agreement that the Cosmo-logical constant, a property of spacetime vacuum, is the source of dark energy. Although theexistence of exotic particles is the main research line in the dark matter issue, quintessencetheories, grounded in some new cosmological field, have the advantage of unifying dark mat-ter and dark energy in a single phenomenon. Vacuum inhomogeneity at a cosmological scaleis suggested as a candidate for quintessence; a brief qualitative analysis of a specific modelwill be presented.

Scalar field fluctuations distorted by two pairs of δ − δ′ interactions

Juan Mateos Guilarte

Universidad de Salamanca (Spain)

Scattering solutions by a double δ − δ′ will be used as one-particle solutions of an scalarfield quantum field theory in a (1 + 1)-dimensional Minkowskian space-time. Following thetrend of mimicking the conducting plates used in a Casimir experiment set-up by means ofpoint δ interactions I will discuss the quantum vacuum interaction between plates relying onthe Lipmann-Schwinger T -matrix.

Supersymmetric partners of the truncated harmonic oscillator

Vicente S. Morales

CINVESTAV-IPN (Mexico)

Supersymmetry transformations of first and second order are used to generate Hamilto-nians with known spectra departing from the harmonic oscillator with an infinite potentialbarrier. Certain systems obtained in a straightforward way through said procedure possessdifferential ladder operators of both types, third and fourth order. Since systems with thiskind of operators are linked with the Painleve IV and Painleve V equations respectively, sev-eral solutions of these non-linear second-order differential equations will be simply found,along with a chain of Backlund transformations connecting such solutions.

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Exotic supersymmetry of reflectionless systems, and solitons

Mikhail Plyushchay

Universidad de Santiago de Chile (Chile)

Finite-gap and reflectionless quantum systems are characterized by exotic supersymme-try. We review the peculiarities of such supersymmetric structure appearing in reflectionlesssystems associated with the soliton and multi-kink-antikink solutions of the KdV and mKdVequations. We also discuss the transmutations of supersymmetry related to the soliton scat-tering.

Some new superintegrable hamiltonian systems, solvable by factorization method

Orlando Ragnisco

Universita di Roma 3 (Italy)

Following a recent paper by Kuru and Negro, we apply the “spectrum generating algebra”method (in the classical case) and the factorisation method (in the quantum case) to solvea pair of new hamiltonian systems, Taub-Nut and Darboux III, which are superintegrabledeformations on curved spaces of the Kepler-Coulomb and Harmonic Oscillator.

Invariant transformations in Euclidean hyperkahler structures

Miguel A. Rodrıguez

Univesidad Complutense de Madrid (Spain)

We present a detailed construction of the invariance algebra of the quaternionic structurein Euclidean spaces. Related results, as maximal subgroups of simple Lie groups and theBerger’s list of holonomy groups, will be also discussed.

The role of indencomposable representations in statistical physics

Yvan Saint–Aubin

Universite de Montreal (Canada)

The transfer matrix is a common tool in statistical physics and shares many propertieswith the Hamiltonian in quantum physics. However, contrarily to Hamiltonians, transfer ma-trices do not need to be Hermitian. They do not even need to have real eigenvalues. Thisdistinction is of physical importance and efforts have been given to understand the representa-tion theory of algebras arising in lattice models like, for example, the families of Temperley-Lieb algebras (TL algebras). One telltale signature of this distinction is the existence of

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Jordan blocks in transfer matrices and, more generally, of indecomposable representations. Ishall give a physical example of these Jordan blocks and report on the efforts to classify allindecomposable representations of the TL algebras and dilute TL algebras.

Haantjes Manifolds and Integrable Systems

Piergiulio Tempesta

Universidad Complutense (Spain)

A general theory of integrable systems is proposed, based on the theory of Haantjesmanifolds. We introduce the notion of symplectic-Haantjes manifold (or ωH manifold), asthe natural setting where the notion of integrability can be formulated. We propose an ap-proach to the separation of variables for classical systems, related to the geometry of Haantjesmanifolds. A special class of coordinates, called Darboux-Haantjes coordinates, will be con-structed from the Haantjes structure associated with an integrable systems. They enable theadditive separation of variables of the Hamilton-Jacobi equation. We also present an ap-plication of our approach to the study of some finite-dimensional integrable models, as theHenon-Heiles systems and a stationary reduction of the KdV hierarchy.

The WKB approximation in deformation quantization

Jaromir Tosiek

Technical University of Lodz (Poland)

An adaptation of the WKB approximation to the deformation quantization is presented.A relationship between the phase σ(~r) of a wave function exp

(i~σ(~r)

)and a respective

Wigner function is derived. Formulas for a Wigner function of a wave function being aproduct of functions and a superposition of functions are proposed. An example of the semi-classical approximation in deformation quantization is analysed.

Position and momentum in the monochromatic Maxwell fish-eye

Kurt Bernardo Wolf

Universidad Nacional Autonoma de Mexico (Mexico)

In geometric optics the Maxwell fish-eye is a medium where light rays follow circles,while in scalar wave optics this medium can only ‘trap’ fields of certain discrete frequencies.In the monochromatic case characterized by a positive integer `, there are 2`+1 independentfields. We identify two bases of functions that serve as wavefields of definite position anddefinite momentum. Their construction uses the stereographic projection of the sphere, andthe identification is corroborated in the `→∞ limit to a homogeneous Helmholtz medium.

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Morning Session, 10 July

10:00- 10:30 Opening Session

Chairman: J.F. Carinena

10:30 - 11:00 V. Aldaya Symmetries from the solution manifold

11:00 - 11:30 M.A. Rodrıguez Invariant transformations in Euclidean hy-perkahler structures

11:30- 12:15 Coffee Break

Chairman: D.J. Fernandez

12:15 - 12:45 A. Bohm Time asymmetric quantum mechanics

12:45 - 13:15 O. Ragnisco Some new superintegrable hamiltonian sys-tems, solvable by factorization method

13:15 - 13:45 Y. Saint-Aubin The role of indencomposable representationsin statistical physics

13:45 - 14:15 M. Plyushchay Exotic supersymmetry of reflectionless sys-tems, and solitons

14:15- 16:00 Lunch

Afternoon Session, 10 July

Chairman: M.A. Rodrıguez

16:00 - 16:30 P.G. Estevez 1+1 spectral problems arising from theManakov-Santini system

16:30 - 17:00 P. Tempesta Haantjes manifolds and integrable systems

17:00 - 17:30 F. Gungor Kac-Moody-Virasoro symmetries of variablecoefficient nonlinear evolution equations in2 + 1 dimensions

17:30- 18:00 Coffee Break

Chairman: A. Ballesteros

18:00 - 18:30 J. Tosiek The WKB approximation in deformationquantization

18:30 - 19:00 M.A. Gonzalez-Leon On the planar Demkov wave functions

19:00 - 19:30 D. Gomez-Ullate Exceptional orthogonal polynomials

19:30 - 20:00 V.S. Morales-Salgado Supersymmetric partners of the truncatedharmonic oscillator

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Morning Session, 11 July

Chairman: M. Santander

09:00 - 9:30 K.B. Wolf Position and momentum in the monochro-matic Maxwell fish-eye

9:30 - 10:00 D.J. Fernandez Penning trap in a rotating magnetic field: co-herent states approach

10:00 - 10:30 V. Hussin Invariant solutions of the supersymmetricCPN−1 sigma model

10:30 - 11:00 N. Gromov Natural limits of electroweak model and con-traction of its gauge group

11:00- 11:45 Coffee Break

Chairman: E. Celeghini

11:45 - 12:15 J. Mateos-Guilarte Scalar field fluctuations distorted by two pairsof δ − δ′ interactions

12:15 - 12:45 C. Lopez Vacuum

12:45 - 13:15 A. Ibort On the structure of Schwinger’s measurementalgebra: groups, groupoids and 2-groupoids

13:15 - 13:45 J.F. Carinena A generalized approach to integrability byquadratures

13:45- 14:00 Closing Session

Wifi Access:SSID: workshopmolmoPassword: mdelolmo

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Participants

Page

Aldaya,Victor valdaya@iaa.es 3Alvarez, Juan J. jjalvarez@infor.uva.esArratia, Oscar oscarr@eis.uva.esAslar, Engin aslarengin@gmail.comBallesteros, Angel angelb@ubu.esBohm, Arno bohm@physics.utexas.edu 3Calzada, Juan A. jacalzada@eii.uva.esCampoamor-Stursberg, Rutwig rutwig@ucm.esCarinena, Jose F. jfc@unizar.es 4Celeghini, Enrico celeghini@fi.infn.itCervero, Jose M. cervero@usal.esCevik, Dogukan cevikdogukan@gmail.comde la Torre, Marina marina@aida.usal.esDıaz-Bautista, Erik ediaz@fis.cinvestav.mxFernandez, David J. david@fis.cinvestav.mx 4Finkel, Federico ffinkel@fis.ucm.esGadella, Manuel manuelgadella1@gmail.comEstevez, Pilar G. pilar@usal.es 4Gomez-Ullate, David dgomezu@ucm.es 5Gomez-Cubillo, Fernando fgcubill@am.uva.esGonzalez Leon, Miguel A. magleon@usal.es 5Gonzalez-Lopez, Artemio artemio@fis.ucm.esGromov, Nikolai gromov@dm.komisc.ru 5Guerrero, Julio juguerre@um.esGungor, Faruk fgungor@dogus.edu.tr 6Herranz, Francisco J. fjherranz@ubu.esHussin, Veronique veronique.hussin@umontreal.ca 6Ibort, Alberto albertoi@math.uc3m.es 6Izquierdo, Jose M. izquierd@fta.uva.esKuru, Sengul sengul.kuru@science.ankara.edu.trLopez-Lacasta, Carlos carlos.lopez@uah.es 7Mateos Guilarte, Juan guilarte@usal.es 7Morales, Vicente S. vmorales@fis.cinvestav.mx 7Munoz, Miguel C. matmcml@ma4.upc.eduNegro, Javier jnegro@fta.uva.esNieto, Luis M. luismi@fta.uva.esOlmo, Mariano A. del olmo@fta.uva.esPascual, Jose F. jfpascua@maf.uva.esPlyushchay, Mikhail m.plyushchay@gmail.com 8Ragnisco, Orlando ragnisco@fis.uniroma3.it 8Rodrıguez, Miguel A. rodrigue@fis.ucm.es 8Saint-Aubin, Yvan saint@crm.umontreal.ca 8Santander, Mariano msn@fta.uva.esTempesta, Piergiulio p.tempesta@fis.ucm.es 9Tosiek, Jaromir tosiek@p.lodz.pl 9Wrickramasekara, Sujeev s-wickram@uiowa.eduWolf, Kurt B. bwolf@fis.unam.mx 9

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