the discretized harmonic oscillator: mathieu functions and a...

t of aquiredct of a

stantlyn andnhar-s forod for

rete,t itatrix.tialg the

of an

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 5 MAY 2003

Downloaded 2

The discretized harmonic oscillator: Mathieu functionsand a new class of generalized Hermite polynomials

M. Aunolaa)

Department of Physics, University of Jyva¨skyla,P.O. Box 35 (YFL), FIN-40014 University of Jyva¨skyla, Finland

~Received 5 August 2002; accepted 21 January 2003!

We present a general, asymptotical solution for the discretized harmonic oscillator.The corresponding Schro¨dinger equation is canonically conjugate to the Mathieudifferential equation, the Schro¨dinger equation of the quantum pendulum. Thus, inaddition to giving an explicit solution for the Hamiltonian of an isolated Josephonjunction or a superconducting single-electron transistor~SSET!, we obtain an as-ymptotical representation of Mathieu functions. We solve the discretized harmonicoscillator by transforming the infinite-dimensional matrix-eigenvalue problem intoan infinite set of algebraic equations which are later shown to be satisfied by theobtained solution. The proposed ansatz defines a new class of generalized Hermitepolynomials which are explicit functions of the coupling parameter and tend toordinary Hermite polynomials in the limit of vanishing coupling constant. Thepolynomials become orthogonal as parts of the eigenvectors of a Hermitian matrixand, consequently, the exponential part of the solution can not be excluded. Wehave conjectured the general structure of the solution, both with respect to thequantum number and the order of the expansion. An explicit proof is given for thethree leading orders of the asymptotical solution and we sketch a proof for theasymptotical convergence of eigenvectors with respect to norm. From a more prac-tical point of view, we can estimate the required effort for improving the knownsolution and the accuracy of the eigenvectors. The applied method can be general-ized in order to accommodate several variables. ©2003 American Institute ofPhysics. @DOI: 10.1063/1.1561156#

I. INTRODUCTION

This paper is closely related to one of the famous eigenvalue problems, namely thaone-dimensional harmonic oscillator. It is common knowledge that if the eigenvectors are reto have continuous second-order derivatives, each eigenvector is expressible as a produHermite polynomial and an exponential term. The corresponding eigenvalues are equidispaced and bounded from below. Another way to state the problem is given by the annihilatiocreation operators which directly diagonalize the Hamiltonian. In comparison, the quartic amonic oscillator was solved by Bender and Wu in Ref. 1. A method for finding eigenvalueanharmonic oscillators was created by Meißner and Steinborn in Ref. 2. A general methpolynomial potentials was introduced recently by Meurice.3,4

Instead of continuous functions, we consider functions defined only on a discequidistantly-spaced and countable set onR. The obvious advantage of this approach is thatransforms the problem into an eigenvalue problem of an infinite-dimensional, tri-diagonal mThe corresponding Schro¨dinger equation is canonically conjugate to the Mathieu differenequation.5 Numerical solutions for noninteger orders are naturally obtained by diagonalizinvery same matrix, see Ref. 6 and the references therein for applications.

In physics, the discretized harmonic oscillator is manifestly realized by the Hamiltonian

a!Electronic mail: [email protected]

19130022-2488/2003/44(5)/1913/24/$20.00 © 2003 American Institute of Physics

8 Dec 2009 to 129.255.1.116. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp

on-be-

xcitedling isonic

Hamil-l.due toe

severalquations. Theurring

e haveof the

rder,tions,

coeffi-ication,n be

icalplingire thermitelation

proofe solu-iven by

s andcases

.ber ofveral

to attice-

. 17.heren op-omialsorderultidi-e

to thele ofuction

1914 J. Math. Phys., Vol. 44, No. 5, May 2003 M. Aunola

Downloaded 2

isolated Josephson junction7,8 and the Hamiltonian of the, slightly misleadingly named, supercducting single-electron transistor~SSET!.8,9 Presently, excited states are seldom consideredcause of the radical approximations under which the Hamiltonian is solved. Even if the estates are numerically obtained, it is not immediately evident, what happens when the coupchanged. In this paper we give an explicit, asymptotical solution for the discretized harmoscillator which corresponds to strong Josephson coupling in case of the SSET. The sametonian also describes the so-called quantum pendulum, or a particle in a periodic potentia10,11

The corresponding asymptotical eigenvalues have been available for almost 50 yearsthe work of Meixner and Scha¨fke on Mathieu functions in Ref. 12. First, by calculating thdeterminant of the matrix representation accurately enough, we continue this expansion byorders in the coupling parameter. Then we propose an ansatz that transforms the matrix einto an infinite set of algebraic equations and proceed by recursively solving these equationgeneral properties of the coefficients in the ansatz can be obtained by studying any occregularities and reinserting these into the solution. Thus, in addition to the eigenvalues, wsuccessfully conjuctured the general form of the asymptotical eigenvectors. In each orderexpansion, the expressions are quoted in terms of an arbitrary quantum number,n, wheneverpossible. The leading terms have been determined and rephrased in terms of an arbitrary om,too. We find that the eigenvectors are asymptotical solutions of certain differential equawhich enables us to obtain further orders in their expansions.

The only real-valued parameter in the solution is the coupling constant, because allcients, both in the eigenvalues and in the ansatz are rational numbers. As a practical applthe rate of convergence of the solution towards numerically obtained, ‘‘exact,’’ solution, careliably estimated. In the asymptotical limit, the dependence in terms ofn andm assumes the formof a simple monomial, at least down to the limits of numerical precision.

The solutions of orderm<5 are very simple to program and directly apply as numersolutions of the discretized harmonic oscillator. For sufficiently small values of the couconstant the eigenvectors are practically exact and thus they facilitate studies which requstructure of the excited states. We have proven, with the help of recursion relations of Hepolynomials, that the first three leading orders of the obtained solution are correct. The calcuup to the seventh order should be performed in the future. We also outline an explicitconcerning the normwise convergence of the eigenvectors. The asymptotical nature of thtions must be stressed. A very thorough introduction on the subject has been given been gBoyd in Ref. 13.

It is justified to ask, is the proposed solution completely new. The answer is, naturally, yeno. Both discretized and discrete harmonic oscillators have been widely studied before. Bothare related to orthogonal polynomials, so the work of Kravchuk14 and Hahn15 must be mentionedThe discrete harmonic oscillator, where the position coordinate is restricted to a finite numvalues, is explicitly solved by Kravchuk polynomials as shown by Lorente in Ref. 16. Sediscretizations of the harmonic oscillator have been previously solved, each giving risespecific class of generalized Hermite polynomials. Discretization by an exponential la$2qn,qnunPZ%, where 0,q,1, defines the so-calledq-deformed harmonic oscillator and generalizedq-Hermite polynomials which are rigorously discussed by Berg and Ruffing in RefFor other applications of theq-deformed harmonic oscillators, see, e.g., Refs. 18 and 19, wother discretizations are reviewed, too. Borzov, in Ref. 20, considers generalized derivatioerators as generators of Hermite polynomials and states that the generalized Hermite polyneither satisfy a second-order differential operator or there is no differential equation of finitefor these polynomials. Many other types of generalizations are also known, see e.g., the mmensional Hermite polynomials of Ro¨sler,21 Hermite polynomials orthogonal with respect to thmeasureujug exp(2j2)dj, whereg.21,22,23 and parabosonic Hermite polynomials.24 In the fu-ture, it must be established whether the presented class of Hermite polynomials is relatedq-Hermite polynomials, if it results from some other discretization or is it an explicit exampthe second group of Borzov’s categorization. Complementary results concerning the introdof distant boundaries for the continuous problem are also known.25,26Finally, it should be empha-


zation,re ex-

plingnd theues ise, e.g.,r, andamined.n as farfor the

equa-ns areiffer-

elp of

r andansatzeneraloeffi-nvec-

theatisfiesnd the

s are

nd/ormetersgiven

ation

1915J. Math. Phys., Vol. 44, No. 5, May 2003 The discretized harmonic oscillator

Downloaded 2

sized that instead of deforming the harmonic oscillator, we solve its common-sense discretiused especially in numerical calculations. The asymptotical effects of the discretization aplicitly calculated.

We also briefly consider the abruptly changing nature of the solutions when the couconstant vanishes. This behavior is evident for both versions of the harmonic oscillator aMathieu differential equation. The asymptotical nature of the solutions and the eigenvalcaused by this divergence. For the Mathieu equation this has been well documented, seRefs. 5, 12, and 27. A more physically motivated approach is given by Bender, PelsteWeissbach in Ref. 28, where, e.g., the instanton equation and the Blasius equation are exThe present methods are closely related to these, although we can not carry the calculatioin the perturbative expansion. This is explained by the necessity of obtaining the expansioneigenvalues which makes the present problem technically more demanding.

The present method can be generalized in a fairly obvious manner. Other differentialtions with analytical solutions can be discretized in the same manner if the correct expansiofound for all parts of the solution. An easier generalization is related to multi-dimensional dence equations with harmonic~quadratic! potential terms. The existing solution29 for Hamiltoniansof one-dimensional arrays of Josephson junctions become more transparent with the hpresent formalism.

The paper is organized as follows. In Sec. II we define the discretized harmonic oscillatoconnect it to the Mathieu differential equation as well as the continuous case. The solutionand the resulting set of equations are reviewed. In Sec. III we quote our conjectures for the gform of the coefficients in the ansatz. We also present the explicit values of the leading ccients. In Sec. IV we study solving the set of equations which yields the asymptotical eigetors. Efficient truncations of the set of equations are explained. The effort for improvingobtained results with the present method is estimated. In Sec. V we prove that the solution sthe difference equations, at least for the three leading orders. The rate of convergence ainduced asymptotical orthonormality are also reviewed. Finally, in Sec. VI the conclusiondrawn and an outlook of future possibilities is given.

A final note for those that are only interested in applying these results in numerical atheoretical analysis. Please review the beginning of Sec. II in order to find the correct parafor the discretized harmonic or Mathieu equation. Then proceed to Sec. III and use theexpressions as approximate solutions in Eq.~30!.

II. THE DISCRETIZED HARMONIC OSCILLATOR

The eigenvalue problem corresponding to the harmonic oscillator is the differential equfor c(x),

21

2

d2c

dx2 1v2x2

2c5lc. ~1!

The eigenvectors corresponding to the well-known eigenvalues,

ln5v~n11/2!, ~2!

wheren50,1,2,..., aregiven by

cn~x!5AnHn~j!e2j2/2. ~3!

Herej5Av x, An is a normalization factor, andHn is the Hermite polynomial of ordern. TheHermite polynomials are solutions of the Hermite differential equation

y922xy812ny50, ~4!


ula,

lation

this

of awhich

rix

SSETtor

thethe

hgorinigen-

for a

s, thet,of


Downloaded 2

wheren50,1,2,... . For our convenience, we write the polynomials, given by Rodrigues’ formas

Hn~j!5~21!n exp~j2!dn

djn exp~2j2!5 (k50

k8

hk(n)jn12(k2k8), ~5!

wherek8ª bn/2c, i.e., k85n/2 if n is even andk85(n21)/2 if n is odd. The quantityk8 provesto be extremely useful in further analysis. The Hermite polynomials satisfy the recursion re

Hn11~j!52jHn~j!22nHn21~j!. ~6!

Many of the generalizations of the Hermite polynomials boil down to a generalization ofrecursion relation.16,17,19,24

The discretized version of Eq.~1! is obtained by restricting the values ofx onto an evenlyspaced, countable subset ofR. This corresponds e.g., to the discretization of charge in caseJosephson junction or a SSET. Only the constant nearest-neighbor coupling is retainedyields a tri-diagonal matrixH(x0) with nonzero matrix elements

H j j ~x0!5 12v

2~ j 2x0!2, H j 11,j~x0!5H j , j 11~x0!52 12. ~7!

Here the parameterx0P@2 12,

12# is the displacement of the origin with respect to the mat

element j 50. All eigenvalues ofH(x0) have been translated by21 in order to simplify thediagonal matrix elements. The standard way to write the Hamiltonian of an inhomogeneousis obtained from Eqs.~7.36! and~7.39! of Ref. 8 and rephrasing it in terms of the number operafor Cooper pairs yields the matrix

H j j(SSET)~N0!5EC~ j 2N0!2, H j 11,j

(SSET)~N0!5H j , j 11(SSET)~x0!52 1

2EJ~u!, ~8!

whereN0 is the number of Cooper pairs which minimizes the charging energy,EC5(2e)2/2CS isthe unit of charging energy, andEJ(u) is the effective Josephson energy which depends ontotal phaseu across the SSET. Consequently, we solve the Hamiltonian of SSET if we findeigenenergies and eigenvector for the discretized harmonic oscillator withv5(2EC /EJ(u))1/2.

In the following, we are searching for eigenvectors with finite Euclidean norm, i.e.,

ici25 (j 52`

`

uc j u2,`. ~9!

The existence and uniqueness of such solutions follows from the generalization of the Gerseigenvalue theory by Shivakumar, Rudraiah, and Williams in Ref. 30. First the number of evalues ofH(x0) on a given interval can be shown to coincide with number of eigenvaluesfinite-dimensional truncation of the matrix,H (N)(x0), if the dimensionN is sufficiently large. Asufficient condition for this is that, in the ordered sequence of diagonal matrix elementdifference between two consecutive values exceeds 43u21/2u52. Furthermore, they prove thafor finite values ofn, the eigenvectorcn

(N) of H (N)(x0) tends to the corresponding eigenvectorH(x0) whenN→`.

We now establish the connection between theH(x0) and the Mathieu differential equation5

d2y

dv2 1~a22q cos~2v !!y50, ~10!

wherea is the eigenvalue, also known as the characteristic value when the solutiony has periodof p or 2p. We follow the derivation of Shirts in Ref. 6 and use Floquet’s theorem to obtain


tion

ian

d

the

theof the

orderathieu

tudiedand

ef. 11.


Downloaded 2

y5exp~ inv !P~v !5exp~ inv !(k

c2k exp~2ikv !, ~11!

where the Fourier expansion ofP(v) has been inserted. This corresponds to the matrix equafor the coefficientsc2k compactly written as

c2k222V2kc2k1c2k1250, ~12!

whereV2k5@a2(n12k)2#/q. This is identical to the discretized harmonic oscillator HamiltonH(x0) with an eigenvaluel after identifications

n522x0 , k5 j , q54/v2, a58l/v2, ~13!

where elements are identified according to 2k↔ j . Thus all results obtained for the discretizeharmonic oscillator also hold for Mathieu functions~11! with parameters given in Eq.~13!. Forx050 andx056 1

2 the solutions ofH(x0) can be chosen to be even or odd with respect toj . Thiscorresponds to writingP(v) in terms of sines and cosines. Special attention must be given toeven solutions ofH(x050), where the resulting equations in the matrix representation read

2c1 /&5l2nc0 , ~14!

2c0 /&1v2c1/22c2/25l2nc1 , ~15!

2c j 21/21v2 j 2c j /22c j 11/25l2nc j , j >2. ~16!

The eigenvalues forx050 correspond to characteristic values$a2n(q),b2n(q)%, while the casex056 1

2 is linked to$a2n11(q),b2n11(q)% as defined in Ref. 5.The asymptotical expansion of the eigenvalues corresponding to the limitq→` or v→0 was

obtained by Meixner and Scha¨fke in Ref. 12. The derivation of the eigenvalues is based onthree-term recurrence relations for the Mathieu functions and the requirement that the normerror in the eigenvalue equation vanishes faster than a specific power ofv. Meixner and Scha¨fkequote the asymptotical characteristic values of the Mathieu equation up to and including thev7 in Theorem 7 in Sec. 2.3. Some error estimates for asymptotical expansions of Mfunctions by Kurz are given in Ref. 27. Because the Mathieu equation is also the Schro¨dingerequation of the quantum pendulum or a particle in a periodic potential, it has been sindependently in physics, too.10,11,31Especially, the same general expansion for eigenvaluesseveral further terms for the ground state energy were obtained by Stone and Reeve in R

In this limit, we can write the eigenvalues ofH(x0) as

ln; (m50

`

ln(m)vm, ~17!

wherev→0, and

ln(m)5 (

k50

m8

ln,k(m)nm12(k2m8) ~18!

with nª2n11 andm85 bm/2c. This structure is identical to that of the Hermite polynomials~5!,if one identifiesn with j. By Ref. 12, the eigenvalues~20! do depend onx0 , but this dependencedecreases exponentially asv→0. The maximal difference is given by12

ln~x056 12!2ln~x050!;~21!nB0~12B1v!v2n23/2exp~28/v!, ~19!

whereB0 andB1 depend onn but not onv.



Downloaded 2

This allows us to write the eigenvalues ofH(x0) as

ln;211vn

22

v2d2

26 2v3d3

211 2v4d4

217 2v5d5

223 2v6d6

227 2v7d7

233 2v8d8

240 2v9d9

247 2v10d10

251

2v11d11

257 2v12d12

261 2v13d13

269 2v14d14

272 2v15d15

279 2v16d16

287 1O~v17!, ~20!

where the coefficientsdk read

d25n211,

d35n313n,

d455n4134n219,

d5533n51410n31405n,

d6563n611260n412943n21486,

d75527n7115617n5169001n3141607n,

d859387n81388780n612845898n414021884n21506979,

d95175045n919702612n71107798166n51288161796n31130610637n,

d105422565n10130315780n81480439190n612135766820n412249346285n21238353840,

d1154194753n111379291385n918186829426n7155529955498n51110241863469n3

141540033277n,

d12510645960n1211187264199n10133678377895n81327725946398n611081358909790n4

1940077055035n2188258370067,

d135440374207n13159495737574n1112155821044201n9128738150160500n7

1144821249264769n51236410740537606n3178243613727607n,

d145578183175n14193209584104n1214215683624295n10174269604367684n8

1537905750769429n611456767306013752n411105711550410653n2

194839535889532,

d15512308013927n1512337227706555n131129437253243675n1112928506455684095n9

129119560960614085n71120372998803922241n51170921920649402745n3

151316344023990085n,

d165530039126159n161117243302735480n1417823093961425652n12

1222043810819026856n1012924952921130025194n8117380315268028265224n6

140851669411526600980n4127983551470330365784n212235152520630714879.


alnownhosen,inant ise

yrsion is

f. 5. Ine.,ssesticaleigen-

up to

q.

l

tg

rmonicmil-

oblem

e

tive


Downloaded 2

We obtain the terms for orders 8<m<11 by exploiting Eq.~18! when explicitly evaluating thedeterminant of Eq.~7!. As a first step, settingx050 halves the dimension of the tri-diagonmatrix. Next, by translating one of the eigenvalues close to zero by substracting the kexpansion of this eigenvalue, the determinant becomes an essentially linear function of the ctranslated eigenvalue. The next unknown term is inserted as a parameter and the determcalculated for several values ofv, preferably in the form$22kv0%k50

3 – 5. This choice lets us separatthe leading correction and the subsequent corrections. In order to obtain the termsd2k andd2k11 ,we must correctly determine all eigenvaluesln whenn<k. Sufficient accuracy is guaranteed busing the high-precision numerics ofMATHEMATICA software. The method for obtaining the ordem.11 requires explicit knowledge on the properties of the eigenvectors and the discusspostponed until the end of Sec. III.

The asymptotical nature of the expansion means that for each value ofv andn, there existsand optimal orderm which minimizes the error in the eigenvalue, i.e., the function

Dl~v,n,m!ªUln2 (m850

m

ln(m8)vm8U , ~21!

with respect tom. The exact eigenvalueln exists and is finite for all nonzero values ofvaccording to the Sturmian theory of second-order linear differential equations, see e.g., Reother words, for sufficiently small values ofv the error is dominated by the first omitted term, i.Dl(v,n,m);uln

(m11)uvm11. Because the asymptotical eigenvalue is divergent, it surely crothe exact eigenvalue whenv is increased, but this occurs outside the range of asymptoconvergence. Similar asymptotical convergence should be observed for the asymptoticalvectors, too. Assumingcn

(m,x0) corresponds to the asymptotical expansion of the eigenvaluesand including ordervm, we expect error in the norm to behave as

icn(m,x0)

2cn(x0)i;C~n,m!vm, ~22!

wherev→0 andC(n,m) is a simple function ofn andm. Although this has not been proven, E~22! appears to be correct and we will ultimately give an approximate expression forC(n,m), too.Outside the regime of asymptotical convergence the error~22! approaches& as the asymptoticasolution becomes orthogonal to the exact one.

Next we show that the discrete eigenvalue problem Eq.~7! is a meaningful asymptotical limiof the continuous harmonic oscillator equation~1!. The problems are identical in the leadininfinitesimal order whenv is infinitesimal, but the limitv→0 is subtle. As long asv.0, both theeigenvalues and eigenvectors of the discretized problem tend to those of the continuous haoscillator with thisv. Forv50 the continuous problem becomes abruptly the free particle Hatonian with solutions

cv50~x!5eikx, lv505k2/2, ~23!

where k is the standard name for the wave number. Simultaneously the discretized prbecomes the well-known nearest-neighbor chain with eigenvectors and eigenvalues,

ck5$eik( j 2x0)% j , lv5052cos~k!. ~24!

For sufficiently small values ofk we havelv50'211k2/2, in agreement with Eq.~23!. Incontrast, we are interested in the bound-state solutions of Eq.~1! and those eigenvectors of thdiscretized problem that can be uniquely related to these continuous solutions forv.0.

The harmonic oscillator is discretized by restricting the values ofx onto a countable andevenly spaced subset ofR. The lowest-order central approximation for a second-order derivais simply


as a

ises a

uerting

d

f

or the

ctions

reads

yields


Downloaded 2

c9~x!5c~x2h!22c~x!1c~x1h!

h2 @1O~h4!#. ~25!

Assumingc(x) to be real analytic allows us to write the numerator of the right-hand sideTaylor series

c~x1h!22c~x!1c~x2h!5 (k51

`2h2k

~2k!!

d2kc~x!

dx2k . ~26!

If h is infinitesimal and as the derivatives ofc are finite in all orders, the only remaining termh2c9(x). Thus, in the lowest infinitesimal order the discretized eigenvalue problem givsecond-order differential equation

21

2

d2cx

dx2 1v2x2

2h2 cx5h22~211l!cx ~27!

which is identical to Eq.~1! apart from the constant2h22 and the redefinitionsv°v/h andl°l/h2. The discreteness of the problem can also be varied by rescaling the value ofv. Thus,instead of decreasing the sizeh of the steps, we seth51 and letv→0. From Eq.~27! we see thatasymptotically the eigenvalues and eigenvectors have the formln;211v(n11/2) and cx

;c(x), as expected.We have already pointed out that the matrixH(x0) in Eq. ~7! can be derived from the Mathie

equation. The underlying reason for this is that the problems are canonically conjugate. Insthe full expansion Eq.~26! into Eq.~27! yields an obvious differential equation incx with respectto x. The canonical transformationid/dx→ v and x→2 id/dv preserves the eigenvalues anproduces the differential equation

2v2

2

d2c v

dv2 2S (k50

`~21!kv2k

~2k!! Dc v5lc v . ~28!

Noticing that the sum is equal to cos(v) and settingvª( v1p)/2, we obtain the canonical form othe Mathieu equation with parameters given in Eq.~13!.

After these important preliminaries, we are able to proceed towards the actual solution fdiscretized harmonic oscillator. In order to treat eigenvectors of all matricesH(x0) on an equalfooting, we replace the indexj by xª j 2x0 . For arbitrary values ofx0 and j the new indexxbecomes a continuous one onR. We thus obtain functionscx

(n) , wheren is the state index. Wepropose that these functionscx

(n) are real-analytic and that they give the eigenvectors ofH(x0)asymptotically, i.e.,

cn(x0)

;$c j 2x0

(n) % j 52`` ~29!

whenv→0. The problem tends to the continuous one in the lowest~infinitesimal! approximationin v. Thus it is reasonable to assume that the lowest-order approximation for the solution funis given bycx

(n);cn(x) asv→0.The general form of the asymptotical solution of the discretized harmonic oscillator now

cx(n)}expS (

k51

`

(l 5k

`

akl(n)v l 21j2kD (

k50

k8

(l 51

`

~hk(n)v l 21bkl

(n)jn12(k2k8)!, ~30!

whereakl(n) and bkl

(n) are constants to be determined. The solution to the continuous casea1,1

(n)521/2 andbk,1(n)51. We are free to normalize the solution so we can chooseb0,l

(n)50 forl .1.


-to ans

sion ofust be

of then.

ofacting

to

ble tod toto show

nd-

r thens

ese ins. This


Downloaded 2

The main point of introducing the functionscx(n) is that they transform the difference

equation-type eigenvalue problem corresponding to the discretized harmonic oscillator ininfinite set of algebraic equations for each value ofn. The eigenvalues~20! appear as parameterand they are required in order to solve the equations for the sets of coefficients$akl

(n)% and$bkl(n)%.

Fortunately, the equations uniquely determine every single coefficient. Because the expanthe eigenvalues is asymptotical, the meaning of the full solution to these equations mdetermined later.

In practice, we need a suitable truncation of Eq.~30! and thus we define an~un-normalized!approximate eigenvector

cn(m,x0)

ª$c j 2x0

(n,m)% j 52`` , ~31!

wherecx(n,m) contains only those terms withl<m. The definition ofcn

(1,x0) obviously coincideswith the continuous solution atx0 . In numerical calculations, and always for even values ofm, wemust truncate the eigenvector with respect toj , by setting (cn

(m,x0)) j50 for componentsu j u. j 0

with a sufficiently large value ofj 0 .We now give the infinite set of algebraic equations corresponding to the transformation

difference equation when the solution functionscx(n) are substituted into the eigenvalue equatio

Rearranging the terms, we find that each equation can be written in the form

cx21(n) 1cx11

(n)

25cx

(n)~2ln1v2x2/2!, ~32!

wherex5 j 2x0 . Inserting the general ansatz~30! into Eq.~32! expresses the equation in termsj andv. The exponential part of the ansatz on the right-hand side canceled simply by substrthe corresponding exponent from those on the left-hand side. This yields an equation

1

2 FexpS (k51

`

(l 5k

`

akl(n)vk1 l 21@~x21!2k2x2k# D (

k50

k8

(l 51

`

~hk(n)v l 21bkl

(n)@Av~x21!#n12(k2k8)!

1expS (k51

`

(l 5k

`

akl(n)vk1 l 21@~x11!2k2x2k# D (

k50

k8

(l 51

`

~hk(n)v l 21bkl

(n)@Av~x11!#n12(k2k8)!G5F2 (

m50

`

ln(m) vm1v2x2

2 G (k50

k8

(l 51

`

~hk(n)v l 21bkl

(n)@Av x#n12(k2k8)!. ~33!

These equations are then expanded as functions ofx andv as the resulting equations are easiersolve. The equations must hold for all values of the linearly independent variablesx andv so eachequation must be solved separately. For the purposes of generality, it would be preferaexpand with respect toj, but the resulting equations are much more difficult, both to obtain ansolve. Nevertheless, the obtained solution can be inserted into to these equations in orderthat the results are correct. This will be done in Sec. V.

In order to obtain the eigenvectorcn(m,x0) we must solve and satisfy all equations correspo

ing to

$$vm8jn12m822l 8% l 850m81k8%m850

m . ~34!

This is of course done recursively, by inserting the known part of the solution and solving fonext level. In order to connect Eq.~34! with the order of the solution, we state that the equatiocorresponding to a fixed value ofm8 uniquely determines the coefficients withl 5m8.

After a while, one starts to see regularities in the coefficients and attempts to express tha functional form. We have been able to find rather general expressions for the coefficient


,that we

al

mmon

d.

rence

rs

nd the

t


Downloaded 2

means that the coefficients have been expressed in terms ofn and the order of the expansionwhenever this is possible. We have conjectured the general form of the terms which meansknow how far away we are from obtaining further terms.

We have found that the functionscx(n,m)(j) are asymptotical solutions of the differenti

equation

S 2 (k50

mvk

~2k!!

d2k

dj2k 1vj2

2 Dcx(n,m)5 (

k50

m

ln(k)vkcx

(n,m) ~35!

in a specific sense. After all derivatives have been carried out, the terms multiplying the coexponential factor cancel up to and including the ordervm. If the solutioncx

(n,m21) is known, weobtain an explicit differential equation for the exponential partvm21f m(j), the correction to theHermite polynomialvm21gm(j) and the energy eigenvalueln

(m) . In case of the ground state anthe first excited state (n51), the condition,gm(j)50 for m.1, renders the problem solvableFor n>2 we must insert the ansatz~30! in order to obtain the solution.

The results are, naturally, in complete agreement with those obtained by using the diffeequation. We are using just another representation of the original problem. Equation~35! enablesus to obtain the solutions for fixed values ofn up to relatively high orders with respect to poweof v. Thus we can both extend the general expression for the eigenenergies in Eq.~20! and thosefor the coefficients in the exponential part of the solutions. For the ground state energy, we fiterms beyond orderv16 to be

2363372562420411197v17

279 26258692522467212813v18

283 2227867608383920243815v19

288

24372199488222446620121v20

292 2352807992522448740907163v21

298

27465886451386334274097895v22

2102 2330752735437897260202410959v23

2107

27654237307570898665851927581v24

2111 21477812451863756884805687589129v25

2118

237132718819258763418452357390369v26

2122

21939848955425261040700592191917783v27

2128

252598573101029275526869814635336865v28

2131

25914101566562517015636997146651378649v29

2137

2172129355454985486683952198830698506149v30

2141

210362392343003738344189045786484697182753v31

2146 1O~v32!. ~36!

The corresponding asymptotical eigenvector contains 31(1131)/25496 linearly independenterms. The coefficient ofv30j2 in the exponential part reads


ated ing

r

In the


Downloaded 2

25207328980459439428858189871778019425519567564728193

2765292404617797269550429065808396826741571584. ~37!

The general solution is given in the next section.

III. THE GENERAL SOLUTION OF THE DISCRETIZED HARMONIC OSCILLATOR

For reasons of completeness and easy accessibility some of the definitions will be repethis section. Themth order solution functioncx

(n,m) , corresponds all terms up to and includinl 5m in Eq. ~30!. The asymptotical expansion of the eigenvaluesln is given in Eq.~20!. The stateindex n determines two expansion parameters,

nª2n11 and k8ª bn/2c, ~38!

where k85n/2 if n is even andk85(n21)/2 if n is odd. The gamma functionG(x) is thegeneralized factorial with the defining propertyxG(x)5G(x11). We need the values for integeand half-integer values which read

G~k!5~k21!!, G~k1 12!522kAp~2k21!!!, ~39!

where the double factorialk!! for integer values ofk is given byk(k22)3¯3(1 or 2). Thecoefficients in the Hermite polynomials simplify to

hk(n)5

~21!k81k22k1(12(21)n)/2n!

~2k1~12~21!n!/2!! ~k82k!!. ~40!

A convenient normalization for the eigenvectors is obtained by requiring that

cx(n);j (12(21)n)/2), x→0, ~41!

i.e., ;1 for even values ofn and;j for odd values ofn. Also bear in mind that

a1,1(n)521/2, bk,1

(n)51, and b0,l (.1)(n) 50. ~42!

Under these constraints we have conjectured that the general form of the coefficients.exponential part,

expS (k51

`

(l 5k

`

akl(n)v l 21j2kD , ~43!

the coefficients can be written as

ak,k1 l(n) 5 (

l 850

l

ak,k1 l[ l 8] nl 8. ~44!

Please note that if the coefficientakl(n) are written as polynomials inn instead ofn, the signs of the

corresponding expansion coefficientsak,k1 l[ l 8] appear to be given by (21)k. An efficient way to

write these coefficients is given by

ak,k1 l(n) 5~21!k222kS (

l 850

b l /2c G~k11/2!Q~k,l ,2k251 l 8!nl 22l 8

G~k1 l !Ap

1 (b( l 21)/2c

Q~k,l ,k221 l 8!nl 2122l 8D , ~45!
l 850

y read


Downloaded 2

whereQ(k,l ,2k251 l 8) and Q(k,l ,k221 l 8) are polynomials ink of orders 2k251 l 8 and k221 l 8, respectively. An important consequence of Eq.~45! is that regardless of the values oflandn we have

limk→`

ak11,k111 l(n) /ak,k1 l

(n) 521/4. ~46!

The explicit expressions for the seven leading coefficients have been obtained and the

akk(n)5

~21!k2222kG~k11/2!

k~2k21!2G~k!Ap,

ak,k11(n) 5~21!k22222kS 1

k1

G~k11/2!

G~k11!Ap

n

kD ,

ak,k12(n) 5~21!k22422kS n1

G~k11/2!

24G~k12!Ap@~3152k140k2!1~9112k!n2# D ,

ak,k13(n) 5~21!k22922kH ~2117k15k2!1~314k!n21

G~k11/2!

24G~k13!Ap3 @~24311119k

11928k211376k31320k4!n1~331101k1104k2132k3!n3#J ,

ak,k14(n) 5~21!k221422kH ~531120k1136k2140k3!n1~37172k132k2!n3/31

G~k11/2!

48G~k14!Ap

3 @~2261292525292132k110675063k2136766856k3140148416k4121300608k5

15544448k61565760k7!/3151~11070160044k1130810k21142112k3181280k4

123168k512560k6!n21~58512288k13585k212696k31960k41128k5!n4#J ,

ak,k15(n) 5~21!k222022kH ~251872672k16580k217684k313164k41452k5!/31~121413744k

14080k211968k31320k4!n21~3451808k1576k21128k3!n4/31G~k11/2!

48G~k15!Ap

3@~74089323013944788389k19627147810k2114943869467k3115287941200k4

110116675072k514238798592k611079918592k71152076288k819052160k9!n/315

1~1825740111037114k127955236k2137919062k3130169312k4114491648k5

14122880k61 636928k7140960k8!n3/31~850501381087k1729798k21752369k3

1447024k41 152576k5127648k612048k7!n5/5#J ,


g to

for

he

h


Downloaded 2

ak,k16(n) 5~21!k222622kH ~3780331496368k1786528k21710816k31339904k4179552k5

17232k6!n/31~697141241312k1303392k21177696k3149408k415120k5!n3/3

1~17217145360k140960k2115360k312048k4!n5/151G~k11/2!

180G~k16!Ap

3@~2246401923868102105728184475128k2155775948330744k2

274654535511116k31 74660144680858k41156803802177352k5

1134434233033760k6170722102090816k71 24590691451392k815680345583616k9

1839668527104k10171921254400k111 2714009600k12!/90091~22093103970

1162201234402k1504160865145k21 882850470198k31986932878421k4

1745434338828k51388089936864k61 138972684672k7133504543744k8

15179637760k91462565376k10118104320k11!n2/211~1520410501991922940k

12784482730k214353707520k314203836660k41 2632731680k511088777440k6

1294912320k7150245120k814874240k91204800k10!n41~2606310112799746k

127798345k2134245070k3126181505k41 12857468k514055200k61792320k7

187040k814096k9!n6#J .

Thus, for an arbitrary orderm, we can obtain the expressions for coefficients correspondin

$vm21j2(m2 l 8)% l 8506 . Furthermore we find

a1,8(n)52~505549159n1177209155n318289645n5140329n7!/237

2~22741702112248825n211518052n4126073n6!/232,

a1,9(n)52~284081902094911419128841068n21221074444682n416195597884n6

121259875n8!/~33247!2~1318785849n1459389255n3129718111n51335617n7!/238,

a2,9(n)5~131257276187n137843099187n311323046497n514456305n7!/244

1~48228434193959845n218787700n41110661n6!/234.

The exponential part~43! is now completely determined up to the ninth order, i.e., knownarbitrary values ofn for terms withl<9.

If we exclude the dependence on 222k and also that given by the gamma functions in tcoefficients, we observe a very distinct regularity. The dependence of the leading powerk in eachpolynomial sequence starting fromn0 for a fixed valuel in ak,k1 l

(n) and going upwards by one botfor l and the power ofn is so far always given by

$a ( l ,l 8)% l 85$a ( l ,0) /~4l 8~ l 8!! !%. ~47!

The initial values in casesl 8<5 are given by

$a ( l ,0)% l 505 5$1,1/4,5/48,5/512,221/96768,113/786432%. ~48!


in Eq.

ingo-re de-rather

is nec-in the

do notld bezov inate on

ntial

ghise that


Downloaded 2

This dependence is by no means proven but it corroborates our choice for the prefactors~45!. A bit surprisingly, we find that the coefficientsak,k1 l

(n) in Eq. ~44! contain information aboutthe energy eigenvalues in casesn50 andn51, i.e.,

a21,m(n50 or 1)52ln50 or 1

(m) . ~49!

The right-hand side is taken from Eq.~20! and it is correctly reproduced form<6.The coefficients$bkl

(n)% determine a set of new polynomials, where the coefficients multiplythe powers ofj depend onv. In the limit v→0 these polynomials tend to the Hermite polynmials. They are, unquestionably, a new class of generalized Hermite polynomials. They afined as parts of the eigenvectors of a Hermitian matrix. Because the exponential part iscomplicated, the measure, with respect to which they become asymptotically orthogonal,essarily a complicated one. It depends on both of the eigenvectors, i.e., it is not a measureclassical sense at all. No simple recursion relation for the polynomials is yet known, and weknow, whether they satisfy any differential equation of finite order. This means that they couan example of the second category of generalized Hermite polynomials as defined by BorRef. 20. Such discussion is beyond the scope of the present study and we will concentrsimpler properties of the polynomials.

Our generalized Hermite polynomials are defined as

Hnv~j!ª(

k50

k8

hk(n)jn12(k2k8), ~50!

where the modified coefficients are given by

hk(n)ªhk

(n)(l 51

`

~v l 21bkl(n)!. ~51!

Because the generalized Hermite polynomialsHnv(j) fix the nodes~zeroes! of the functionscx

(n) ,it is equally important to obtain correct polynomials as it is to obtain the correct exponefactors.

We conjecture that the general form of the coefficients$bkl(n)% reads

bkl(n)5 (

l 850

l 21 S (l 51

2(l 21)2 l 8

~rl 8 l

( l )1@~12~21!n!/2#r

l 8 l

( l )!kl D ~k8! l 8, ~52!

whererl 8 l

( l )and r

l 8 l

( l )are constants. Additionally,r

l 8 l

( l )50 when l 52(l 21)2 l 8 or l 85 l 21. This

expansion with respect tok and k8 shows that even and odd values ofn should be treatedseparately.

Some general properties of the coefficientsrl 8 l

( l )and r

l 8 l

( l )have been gleaned. The recurrin

appearance of the factor (10k82k) is by far the most striking of the observed regularities. Tfactor may, in time, explain some properties generalized Hermite polynomials. We conjectur

(l 850

l 21

~r l 8,2(l 21)2m02 l 8( l ) k2(l 21)2m02 l 8nl 8!5kl 212m0~10k82k! l 2122m0P~2m0 ,l !, ~53!

where 2m0, l and P(2m0 ,l ) denotes a (2m0)th order polynomial ink and k8. Similarly, thedifference between even and odd values ofn corresponds to

(l 850

l 22

~ r l 8,2l 232m02 l 8( l ) k2l 232m02 l 8nl 8!5kl 212m0~10k82k! l 2222m0P~2m0 ,l !, ~54!


e

c-

e


Downloaded 2

where 2m0, l 21 andP(2m0 ,l ) again denotes a (2m0)th order polynomial ink andk8.In the leading and next-to-leading orders the polynomialsP(0,l ), P(2,l ), P(0,l ), andP(2,l )

have been explicitly evaluated. Thus, we define the quantities

B~ l ,2l 22!ªkl 21~10k82k! l 21

48l 21~ l 21!!~55!

and

B~ l ,2l 23!ªkl 22~10k82k! l 23

5348l 21~ l 22!!@~ l 22!658~k8!21~4022126l !k8k1~8l 231!k2#, ~56!

which has been confirmed up to the sixth order, i.e.,l 56. Similarly, the leading differences givrise to the quantities

B~ l ,2l 23!ª4kl 21~10k82k! l 22

48l 22~ l 22!!~57!

and

B~ l ,2l 24!ªkl 22~10k82k! l 24

5348l 21~ l 23!!@~2632l 22576!~k8!21~14702504l !k8k1~32l 2145!k2#.

~58!

The leading terms are very similar, but also the next-to-leading termsB( l ,2l 23) and B( l ,2l24) share several common features. Most importantly, thel -dependence in the polynomial setion is identical, apart from a factor of 4.

Below, we give the explicit values of the coefficientsbkl(n) in cases 2< l<7. It is convenient to

separate the even and odd values ofn, because the correct expansion parameter appears to bk8.Please note that these expressions automatically yieldb0,l

(n)50 for l .1,

bk,2(2k8)5@~3k2k2!1~10k!k8#/48,

bk,2(2k811)5bk,2

(2k8)1k/12,

bk,3(2k8)5@~855k264k2214k315k4!1~784k148k22100k3!k81~1316k1500k2!~k8!2#/23040,

bk,3(2k811)5bk,3

(2k8)1@~249k149k2220k3!1~532k1200k2!k8#/11520,

bk,4(2k8)5B~4,6!1B~4,5!1@~371385k2203498k2212129k311438k4!1~1110698k1102042k2

226252k3!k81~496932k193984k2!~k8!21560200k~k8!3#/23224320,

bk,4(2k811)5bk,4

(2k8)1B~4,5!1B~4,4!1@~67680k112347k222602k3!1~108544k133762k2!k8

1114456k~k8!2#/3870720,

bk,5(2k8)5B~5,8!1B~5,7!1@~278751375k2202014918k2135222268k314026748k4128158k5

29944k6!1~713250468k2281790420k2261452368k32196176k41209856k5!k8

1~1105743252k1198178852k2210630680k32275344k4!~k8!21~319197168k

181282336k2222799744k3!~k8!31~271672512k1148408976k2!~k8!4#/22295347200,



Downloaded 2

bk,5(2k811)5bk,5

(2k8)1B~5,7!1B~5,6!1@~119817225k223468037k2212060122k32330312k4

1100198k5!1 ~436319556k1103769756k225886336k321921112k4!k8

1~321608148k1118383396k22904080k3!~k8!21~224147856k1120486304k2!

3~k8!3#/11147673600,

bk,6(2k8)5B~6,10!1B~6,9!1@~134035780725k2166751340588k2139327194883k3

22269605874k42477614210k5219226552k61221782k71484k8!1~413990823078k

2217584747090k2122678956764k318841166604k41479019924k526549884k6

1566984k7!k81~526339688532k2155591533528k2255003198072k323518713436k4

185514000k5224903296k6!~k8!21~556945898088k1131085561976k2

12790556248k32280060176k41424040144k5!~k8!31~116760015552k

134523271136k225865150192k323338174576k4!~k8!41~79966766400k

146102886720k2110162787360k3!~k8!5#/11771943321600,

bk,6(2k811)5bk,6

(2k8)1B~6,9!1B~6,8!1@~34460588160k226910050283k2172069996k3

11282383895k41103465570k52948002k62316976k7!1~216801198648k

221671791146k2218471533106k321699322576k4178651584k515865684k6!k8

1~345295895928k196181762100k211478206984k321409258180k4160822872k5!

3~k8!21~160052617776k164547633160k213204992824k321898232512k4!~k8!3

1~83156900448k147130830560k2110276562912k3!~k8!4#/5885971660800,

bk,7(2k8)5B~7,12!1B~7,11!1@~21167446950775125k234318046368345140k2

113674300462898392k321352901404372446k422843855572731k5

111311875159790k61704407032828k7112949326156k82177366189k9

137677640k10!1~59570630372492640k260644270495554704k2

110066261151648252k31104602336760652k42246415137367020k5

220207362771548k62460168946016k712604105504k822483040560k9!k8

1~99669485611466412k239020273844707836k212200698814542984k3

120707130729546001212428368788100k515622413614220k6182814211480k7

170784553840k8!~k8!21~82488078028378080k218442822328400480k2

29100818756007520k32964844434165920k4224156442527360k523013682511840k6

21116228072960k7!~k8!31~66588038149135200k118345507366303440k2

11191268975557840k3110518809509520k4142474114642960k5

110244315921840k6!~k8!41~10839030004200960k13516288982521792k2

2363895953410496k32304221200739456k4251610667908800k5!~k8!5


pres-

glength

y exactre-

te ex-all-

to,

ate liketicale

nt


Downloaded 2

1~6218212960526208k13705496740373376k21898601964676416k3

1110684037464000k4!~k8!6#/1542595452862464000,

bk,7(2k811)5bk,7

(2k8)1B~7,11!1B~7,10!1@~2460686821541975k21941941074537755k2

1366877584331212k3141494582964306k429965970910165k521112954021925k6

234972438722k71958101144k813798795k9!1~9145976126266080k

23823250702059872k22191143081971676k31173415983605096k4

124948282593576k51815890000796k6241300603344k711974092120k8!k8

1~19045045703842332k2607795420829248k221422456568355676k3

2195873935369616k422655255816252k51611948653316k6298001423520k7!

3~k8!21~18941405236672032k15863460843089248k21345511721120640k3

255753844615456k42654723253184k511792716525600k6!~k8!3

1~6296099301099168k12692631923242160k21232744611197904k3

259049744898144k4214606140268720k5!~k8!41~2605202959125888k

11525593370591680k21365590616623232k3144670372947200k4!

3~k8!5#/257099242143744000.

In combination with the exponential parts these coefficients determine explicit, analytical exsions for solution functioncx

(n,7) for arbitrary values ofn.32

It must be re-emphasized that Eq.~30! is an asymptotical solution. Two partially overlappinreasons for this behavior must be stated. First, the solution depends on two independentscales, i.e.,x andv, and second, the coordinate transformationx°j5Avx is singular atv50.These points are rather extensively covered in Ref. 13. The eigenvalues are asymptoticallfor even values ofn at x050 and, probably, for odd values ofn at x056 1

2. Because the errodecays exponentially in 1/v, this dependence onx0 vanishes much before the asymptotical bhavior of Eq.~22!, i.e., icn

(m,x0)2cn

(x0)i;C(n,m)vm, appears.Comparison against numerically obtained eigenstates allows us to give an approxima

pression for the functionC(n,m). The validity of the calculations is limited by the numericprecision, i.e., to norms of the order of 10211– 10212. We have employed the reliable diagonaization routines ofMATLAB software for this purpose. We have studied eigenvectors upn'40– 50 and the corresponding asymptotical solutionscn

(m,x0) up to the fifth order. A reasonableorder-of-magnitude estimate for the error in the Euclidean norm, whenn<40, is given by

C~n,m!'cmn2m, ~59!

where

c150.03, c250.002, c350.0006, c451.531026, and c55331028. ~60!

There is a slight difference between even and odd cases, but this is insignificant in an estimthis. The value ofc5 is set to fit the observed trend in the other coefficients as the asymptobehavior is only glimpsed. In casesm52 andm54, it is vitally important to remember to truncatthe asymptotical eigenvectorcn

(m,x0) correctly.For larger values ofn, one needs very small values ofv in order to obtain accurate or eve

reasonable results. But for relatively small values ofn, sayn<10, the error is extremely small a


igheymp-

-order

usex-

rder

envec-on andare.

.o arbi-

s

d also

er, we


Downloaded 2

v'0.01. The strong dependence onn means that the first few states can be obtained to a hprecision even for quite strong couplings in the neighborhood ofv'0.1. We have determined thground staten50 up to the 31st order and numerical comparison strongly supports the astotical behaviorvm for m<13.

In order to make the above discussion more concrete, we explicitly give the secondsolutions as functions ofv, j5Av x, n ~not n) andk8. For even values ofn we find the solutionfunction

cx(n,2)5An,x0

exp~2~ 121~312n!v/32!j21~v/96!j4!(

k50

k8

~hk(n)j2k~11~3k2k2110kk8!v/48!!,

~61!

whereAn,x0is a normalization factor which ensures thaticn

(2,x0)i51. For odd values ofn theresult is nearly identical, i.e.,

cx(n,2)5An,x0

exp~2~ 121~312n!v/32!j21~v/96!j4!(

k50

k8

~hk(n)j2k~11~7k2k2110kk8!v/48!!.

~62!

The tiny difference 3k→7k in the generalized Hermite polynomial is very important, becaotherwise the asymptotical convergenceicn

(m,x0)2cn

(x0)i;v2 does not appear. The common eponential part in the third order solution functioncx

(n,3) reads

exp~2~ 121~312n!v/321~53169n121n2!v2/1536!j21~v/961~1116n!v2/1024!j4

2~v2/1280!j6!). ~63!

The explicit solution functioncx(n,m) solves the asymptotical eigenvalue equation up to the o

vm and yields a normwise convergence of;vm.When employing these asymptotical solutions, one should first study, how accurate eig

tors are required for the problem at hand. The next step is to choose the order of the solutithe correct truncation with respect tox. Then, the calculations are performed and the resultsobtained, hopefully faster than with the conventional approach of numerical diagonalization6

IV. COMMENTS ON SOLVING THE ANSATZ

In this section we discuss how to solve the set of algebraic equations resulting from Eq~33!as effectively as possible. First we observe that the zeroth order, i.e., terms proportional ttrary powers ofj are satisfied by the fact exp(0)51. Next, all equations related to terms

$vjn1222l 8% l 85011k8 ~64!

are identically satisfied because of the recursion relation~6! rewritten in terms of the coefficienthk

(n) . A careful reader notices that terms proportional tobk,l 52(n) do appear, but they identically

cancel and thus they are not constrained in this order.From here on, we proceed by recursively solving the coefficients for the next order an

for sufficiently many values ofn so that all coefficients in the expansions of$akl(n)% and $bkl

(n)%have been constrained. In reality, we first obtained the solution functioncx

(n50,m56) and a poorlyformulated expression for arbitrary second-order solution, i.e.,cx

(n,m52) , but let us proceed in theway this should be done. Because the equation are quite difficult to handle with pen and papchose to write and simplify the equations withMATHEMATICA software.33

We first consider the casesn50 andn52 as simple examples. Forn50 we expand Eq.~33!up to and including orderv3 to find


theee with

r

n-

ding to


Downloaded 2

H 12v

21v2S a1,2

(0)11

81

x2

2 D1v3F21

481

a1,2(0)

21a1,31a2,2

(0)1x2S 21

422a1,2

(0)16a2,2(0)D G J

5F12v

21

v2~x211/16!

21

v3

512G . ~65!

Immediately, we obtain

a1,2(0)523/32 anda2,2

(0)51/96. ~66!

Inserting these into Eq.~65! givesa1,3(0)5253/1536.

In the casen52 and we examine all terms below the order ofv4. The generalized Hermitepolynomial now reads

H2v~j!52214vx2~11b1,1

(2)v1b1,2(2)v21b1,3

(2)v3!1O~v5!. ~67!

Expanding all terms and moving them onto the same side yields the equation

05v2S 23

321a1,2

(2)22b1,1(2)D1v3F2

521

15362

5a1,2(2)

21a1,3

(2)1a2,2(2)1b1,1

(2)22b1,2(2)

1x2S 243

16212a1,2

(2)16a2,2(2)D G1v4F 341

245761

~a1,2(2)!2

22

5a1,3(2)

21a1,4

(2)1a1,2(2)S 9

822b1,1

(2)D2

b1,1(2)

41b1,2

(2)22b1,3(2)1x2S 953

76812~a1,2

(2)!2212a1,3(2)237a2,2

(2)16a2,3(2)1a1,2

(2)S 29

2210b1,1

(2)D2

39b1,1(2)

16 D 1x4S 29

2414a1,2

(2)232a1,3(2)D G . ~68!

Notice that all terms proportional tov0 and v1 have canceled out, which again shows thatlowest-order approximation for the eigenvalue and eigenstate are already correct and agrthe results for the continuous case. The three coefficients related to thec (2,2) can be solved fromthe coefficients ofv2, v3x2, andv4x4 and they read

a1,2(2)527/32, a2,2

(2)51/96, andb1,1(2)51/4. ~69!

Substituting these into the set of equations and extending the calculation to orderv6 we find thesubsequent coefficients to be

a1,3(2)52275/1536, a2,3

(2)523/1024, a3,3(2)521/1280, andb1,2

(2)537/256. ~70!

After solving a sufficient number of coefficientsakl(n) and bkl

(n) one should start searching foregularities in the solution.

Almost immediately we guessed the polynomial character ofakl(n) , first in terms ofn and later

noticing that they should be written in terms ofn as in Eq.~44!. This considerably helps solvingthe coefficientsbkl

(n) as for larger values ofn the coefficientsakl(n) appear as constants, not u

knowns.In the beginning, we tried to solve all possible terms up to a given order inv. First one should

notice that only terms withl<m are required for the solution functioncx(n,m) . Assuming that the

previous orders have been explicitly obtained, means that only the equations corresponm85m in Eq. ~34! have to be solved. In addition, generally known coefficientsak,m

(n) identicallysatisfy equations corresponding to the highest powers ofj. Explicitly, if we assume that coeffi-cients$ak,m

(n) %k5k0

m are known, only the equations for


er

g theeveral

ry timeit

omial

asythese

rrectthe

n

n-

olution-lren-

p to

ationsmuchle that

d there, as the

, if atsolid

or the

one.

anded for


Downloaded 2

$vmjn12k022l 8% l 850

k01k8~71!

are required and the expansion of Eq.~33! has to be carried out up to and including the ordvm1k01n/2 for coefficientsl<m.

After obtaining a rather complicated expression for the coefficientsbk,l 53(n) , we happened to

transform it into form equivalent to the present form and conjecture the general form ofbkl(n) in Eq.

~52!. The most important lesson taught by the discretized harmonic oscillator when solvincoefficients is that your numbers may be wrong, but the general forms usually are not. On soccasions, this became painfully obvious when the numbers did not check. Each and evethe general forms were correct, but the used expansion of Eq.~33! or the numbers inserted intowere not.

Later on, we started to study the regularities in the general expressions. The polyn

structure of the coefficientsakl[ l 8] that do not contain any gamma functions was relatively e

obtain, but the other set required a real stroke of luck. We managed to write some of

coefficientsakl[ l 8] as explicit products. After being pointed out, byMATHEMATICA , that the first two

could be written in terms of gamma functions, it was only a question of finding the cogammas before Eq.~45! was written. In order to appreciate the technical part of obtaininggeneral form of the coefficients we point out that the coefficientak,k14

(n) was completed by solvingthe 12th order solutioncn50

(m512,x0) and confirmed by the casen51. Further terms have beeobtained by solving the asymptotical differential equations~35!.

The regularities in the coefficients$bkl(n)% have been found out using by studying the expa

sions with respect tok andk8. By conjecturing the recurring appearance of (10k82k) in Eqs.~53!and~54! it becomes possible to solve the quantities defined in Eqs.~55!–~58!. In addition to these,the general expression forrk,l 23

( l ) can be obtained from the known coefficients.Finally, we will estimate the difficulty of obtaining the explicit asymptotical solutioncn

(m,x0) .We assume that both the expansion of the eigenvalues up to the required order and the scn

(m21,x0) have been obtained in advance. The coefficientsakm(n) can be determined from the expo

nential parts of the eigenvectors up to and including the casen5m21. The completely generaexpressions in Eq.~45! are finished at much slower a pace. The asymptotically satisfied diffetial equations~35! speed up this process considerably.

Obtaining the coefficientsbkm(n) is more difficult. The general form~52! shows that all states up

to n54m23 must be solved. The explicit expressions for the leading parts, i.e.,B( l ,2l 22),B( l ,2l 23), B( l ,2l 23), andB( l ,2l 24) make this task easier by five states. Thus all states un54l 28 must be found, unless further general properties are found.

Regardless of these simplifications, the number of required terms and participating equgrows quite fast. Obviously, the general form of the coefficients in the exponential factor iseasier to obtain and thus they should be applied as early as possible. It is also possibconsiderable simplifications or generalizations for the known coefficients lurk just arouncorner. This has already happened on several occasions so far. We still choose to pause hegiven general expressions have been validated rather convincingly and it not obvious, howall, the next orders in the expansion would improve the results qualitatively. We hope afoundation has been laid for those striving towards the complete, asymptotical solution fdiscretized harmonic oscillator.

V. PROVING THE SOLUTION AND SOME GENERAL PROPERTIES

Finally, we attack the difficult problem of actually showing that the solution is a generalThus far we have solved the equations for an increasing number of eigenstates using Eq.~33!. Thisformulation is the best if actual numerical values of the coefficientsakl

(n) andbkl(n) are sought after.

This is explained by symbolic math being most effective when the number of unknownssymbols is as small as possible. In principle, the process explained below could be us


he full

e can

orderctionsbyantnt

rsion


Downloaded 2

obtaining recursion relations between the coefficients of the solution and, subsequently, tsolution. Presently, we only show that the equations corresponding to leading orders up tov3 aresatisfied identically.

We have to solve the equations corresponding to$vmjn1m22l 8% l 850k81m in order to obtain the

mth order solution. We have now obtained the explicit solution up to the seventh order so wcheck if it is correct. For this purpose, we must write Eq.~33! explicitly in terms ofuªAv andj, although odd powers ofu eventually cancel. Multipliers ofakl

(n) andbkl(n) now read

u2(l 21)@~j6u!2k2j2k# and ~j6u!n12(k2k8), ~72!

respectively. On the right-hand side the nontrivial term is given byvj2/2. Expanding all termsmultiplying a fixed termhk

(n) up to the order yields terms

hk(n)F11u2S 2~k2k8!2n/21

j2

21

~n2112~k2k8!!~n12~k2k8!!

j2 D G5hk

(n)S 11u2S 2n/21j2

2 D D . ~73!

The terms proportional toj2 cancel and equating each power ofj separately yields an equation

hk(n)2~k82k!1@2~k11!27~k11!#hk11

(n) 50, ~74!

where the signs1 and 2 corresponds the even and odd values ofn, respectively. The aboveequation is identically satisfied by the Hermite polynomials, which proves that the first-solutioncn

(1,x0) is correct. A careful observer immediately asks about the second order correbk,2

(n) which also yield terms proportional tou2. However, these coefficients are not fixed at allEq. ~33! in the orderu2. The only term that is easily solvable from this relation in the domincoefficienta1,1

(n)521/2 which removeshk21(n) from the recursion relations. Later on, the domina

coefficients$akk(n)%k51

m cancel the termhk2m(n) in the ordervm.

In the next orderv2 we insert the solved coefficients and obtain for even values of a recurelation

6~n1222k!hk21(n) 1@~2k31k2~42211n!26n23n22k~2619n25n2!#hk

(n)2~11k!~112k!

3~22131k1k225n25kn!hk11(n) 12~2k14!~2k13!~2k12!~2k11!hk12

(n) 50, ~75!

which is again identically satisfied by the Hermite polynomials. For odd values ofn, we find asimilar recursion relation, once we replacek85n/2 by k85(n21)/2. This completes the proof inorderv2 and validates the second-order eigenvectorscn

(2,x0) .In the third order the recursion relation for even values ofn reads

180~2412k2n!hk22(n) 130~62242k224k214k3174n210kn222k2n117n2110kn2!hk21

(n)

1@2450n2450n2290n3210k51k4~24521105n!1k3~2233212458n2300n2!

1k2~423014912n21204n21125n3!1k~23002585n21258n21179n3!#hk(n)

1~11k!~112k!~102212705k13684k21326k315k422274n24430kn21726k2n

250k3n1454n21 579kn21125k2n2!hk11(n) 216~11k!~21k!~112k!~312k!~1101101k

15k2250n225kn!hk12(n) 1256~11k!~21k!~31k!~112k!~312k!~512k!hk13

(n) 50.

~76!


s of

atrix

for

firstvectorf

atr

-o

eresult


Downloaded 2

Because the Hermite polynomials satisfy this and the corresponding relation for odd valuenthe solutioncn

(3,x0) has been rigorously proven as correct.

The eigenvectorscn(m,x0) tend to the eigenvectorscn

(x0) of H(x0) at an asymptotical rateproportional tovm. The exact eigenvectors are orthogonal as eigenvectors of a Hermitian mand by their closure relation we can write

cn(m,x0)

;cn(x0)

1vm(n8

bn8cn8

(x0), ~77!

wherebn8 are finite constants such that(nubnu2,` in the limit v→0. The orthonormality relationfor the asymptotical solutions thus reads

^cn(m,x0)uc

n8

(m,x0)&5dnn81O~vm!, ~78!

provided that the sum(nubnu is finite for both states. In other words, the eigenvectorscn(m,x0)

become orthonormal at the rate ofvm. Numerical checks seem to confirm this, at leastrelatively small values ofn.

As a final effort, we outline a plausible ‘‘proof’’ for the asymptotical convergence. As astep, we show that without loss of generality we can examine a finite truncation of the eigencn

(n,x0) , the vectorcn, j 0

(x0)ª$c j 2x0

(n) % j 52 j 0

j 0 for sufficiently largej 0 . For sufficiently large values o

u j u the eigenvalueln becomes insignificant in Eq.~32! and we write an approximate equation

~cx21(n) 22cx

(n)1cx11(n) !/~x2!5v2. ~79!

For sufficiently large values ofx and/orj the sign ofcx(n) is constant and this equation shows th

the functioncx5c j 02x0

(n) exp(2v(x22j02)/(21«)), for some small«.0, is a dominant sequence fo

cx(n) . Now, the limiting sequence of norms

limj 0→`

icn(x0)

2cn, j 0

(x0)i ~80!

vanishes exponentially with respect toj 0 . In other words, we can always find a finitej 0 such thatthe error in the norm is sufficiently small.

Next we use the fact that solutioncn(m,x0) satisfies the eigenvalue equation~32! up to the order

vm when written in terms ofj. Thus we can write

cx21(n,m)1cx11

(n,m)

2cx(n,m)~2ln1vj2/2!

511O~vm11!. ~81!

We fix the scales of the eigenvectors by setting (cn(m,x0)) j5(cn

(x0)) j for an arbitraryj . It would be

very tempting to say that Eq.~81! implies (cn(m,x0)) j 115(cn

(x0)) j 11(11O(vm11)) and then won-der why convergence is not asymptotically proportional tovm11. As already explained the solution cn

(m,x0) does not fix the coefficientsbk,m11(n) which most definitely yield terms proportional t

vm. Thus we obtain a relation

~cn(m,x0)

! j 115~cn(x0)

! j 11~11O~vm!!. ~82!

By matching the eigenvectors atj 50, and expanding the components to the finite values6 j 0

shows that the order of error isvm for all components withu j u< j 0 . Becase the error caused by thtruncation is insignificant the result holds for the full eigenvectors and we obtain the desired


lator.ified rateccord-

multa-

ations

dimen-nt andto a

nslimit.

e form.

f Ex-at

thor

or

da-


Downloaded 2

icn(m,x0)

2cn(x0)i;vm, ~83!

or at least show that the result is quite plausible.

VI. CONCLUSIONS

We have obtained an explicit, asymptotical solution for the discretized harmonic oscilBoth the eigenvalues and eigenvectors have been obtained and we can choose a prespecof convergence towards the exact solutions. This is done by truncating the ansatz solution aingly. Because the problem can be mapped onto the Mathieu differential equation, we sineously provide asymptotical expressions for the Mathieu functions. The Schro¨dinger equation ofthe quantum pendulum corresponds to the Mathieu equation, which yields immediate applicfor the results.

The method described above can be generalized to accommodate several coordinatesions with only minor changes. This should make the results of Ref. 29 both more transparemore rigorous. The tunnelling–charging Hamiltonian of a Cooper pair pump correspondsmodified multidimensional Mathieu equation.

Alternatively, ansatzes similar to Eq.~30! could be constructed in case of difference equatiothat become identical to analytically solvable differential equations in some asymptoticalInitially, the problem assumes the form of an infinite-dimensional, two-parameter~eigenvalue!problem, where the asymptotical solutions~eigenvalues and eigenvectors! must be obtained. Theansatz maps the problem onto an infinite set of algebraic equations that must solved. If thof the ansatz is correct, one may determine some general properties of the exact solution

ACKNOWLEDGMENTS

This work has been supported by the Academy of Finland under the Finnish Center ocellence Programme 2000–2005~Project No. 44875, Nuclear and Condensed Matter ProgramJYFL!. Dr. L. Kahanpa¨a is acknowledged for insightful discussions and suggestions. The authanks Professor J. Timonen for valuable references added to the final draft of the paper.

1C. Bender and T. T. Wu, Phys. Rev.184, 1231~1969!.2H. Meißner and E. O. Steinborn, Phys. Rev. A56, 1189~1997!.3Y. Meurice, quant-ph/0202047.4B. Bacus, Y. Meurice, and A. Soemadi, J. Phys. A28, L381 ~1995!.5M. Abramowitz and I. A. Stegun, inHandbook of Mathematical Functions~Dover, New York, 1966!, Chap. 20.6R. B. Shirts, ACM Trans. Math. Softw.19, 377 ~1993!.7E. Celeghini, L. Faoro, and M. Rasetti, Phys. Rev. B62, 3054~2000!.8M. Tinkham, in Introduction to Superconductivity, 2nd ed.~McGraw-Hill, New York, 1996!, pp. 257–277.9T. M. Eiles and J. M. Martinis, Phys. Rev. B50, R627~1994!.

10J. Dorignac and S. Flach, Phys. Rev. B65, 214305~2002!.11M. Stone and J. Reeve, Phys. Rev. D18, 4746~1978!.12J. Meixner and F. W. Scha¨fke, ‘‘Mathieusche funktionen und spha¨roidfunktionen,’’ Die Grundlehren der Mathematische

Wissenschaften~Springer-Verlag, Berlin–Go¨ttingen–Heidelberg, 1954!, Vol. 71. A more accessible textbook by ProfessMcLachlan was published originally in 1947. N. W. McLachlan,Theory and Application of Mathieu Functions,coor. ed.~Dover, New York, 1964!.

13J. P. Boyd, Acta Applic.56, 1 ~1999!.14M. Krawtchouk, C. R. Acad. Sci. URSS, Ser. A189, 620 ~1929!.15W. Hahn, Math. Nach.2, 4 ~1949!.16M. Lorente, Phys. Lett. A285, 119 ~2001!.17C. Berg and A. Ruffing, Commun. Math. Phys.223, 29 ~2001!.18G. Parisi, J. Phys. A27, 7555~1995!.19D. Bonatsos, C. Daskaloyannis, D. Ellinas, and A. Faessler, Phys. Lett. B331, 150 ~1994!.20V. V. Borzov, math.QA/0101216.21M. Rosler, Commun. Math. Phys.192, 519 ~1998!.22M. Rosenbaum, math.CA/9307224, 1993; SIAM OP-SF 25.23H. Dette, AMS Proc.348, 691 ~1996!.24S. Jing and W. Yang, math-ph/0212011.25G. Barton, A. J. Bray, and A. J. McKane, Am. J. Phys.58, 751 ~1990!.26H. I. Elim, quant-ph/9901009.27J. Meixner, F. W. Scha¨fke, and G. Wolf,Mathieu Functions and Spheroidal Functions and their Mathematical Foun

tions, Lecture notes in Mathematics, Vol. 837~Springer-Verlag, Berlin–Heidelberg–New York, 1980!.


jyu.fi/given up


Downloaded 2

28C. M. Bender, A. Pelster, and F. Weissbach, J. Math. Phys.43, 4202~2002!.29M. Aunola, cond-mat/0206507, 2002.30P. Shivakumar, R. Sharpley, and N. Rudraiah, Linear Algebr. Appl.96, 35 ~1987!.31R. B. Dingle and H. J. W. Mu¨ller, J. Reine Angew. Math.211, 11 ~1962!.32Matlab m-files for reconstruction of the wave functions~up to fourth or fifth order! are available at http://www.cc.jyu.fi/

;mimaau/harmonic33Many MATHEMATICA notebooks containing much of the data used in calculations is available at http://www.cc.

;mimaau/harmonic. The general results have been compiled into Mathieunewgen.nb, where the eigenvalue isto 19th order as a well as the general expression for the coefficientak,k17

(n) .


the discretized harmonic oscillator: mathieu functions and a...

Documents