Approximate solutions to the quantum problem of two apposite charges in a constant magnetic field

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Approximate

solutions

to

the

quantum

problem

of

two

opposite

charges

in

a

constant

magnetic

field

J.S. Ardenghi

a

,

M. Gadella

b

,

c

,

∗

,

J. Negro

b

a_{IFISUR,DepartamentodeFísica(UNS-CONICET),AvenidaAlem1253,BahíaBlanca,BuenosAires,Argentina} b_{DepartmentofTheoretical,AtomicPhysicsandOpticsandIMUVA,UniversityofValladolid,47011Valladolid,Spain} c_{GrinnellCollege,DepartmentofPhysics,Grinnell,50112IA,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received23November2015

Receivedinrevisedform23March2016 Accepted24March2016

Availableonline29March2016 CommunicatedbyP.R.Holland

Keywords: Integrablesystems Razavyequation Hillequations Perturbativesolutions

We consider two particlesof equal mass and oppositecharge in aplane subject to aperpendicular constantmagneticfield.Thissystemisintegrablebutnotsuperintegrable. Fromthequantum pointof view,thesolutionisgivenbytwofourthdegreeHilldifferentialequationswhichinvolvetheenergyas wellasasecondconstantofmotion.Therearetwosolvableapproximationsinrelationtothevalueofa parameter.Startingfromeachoftheseapproximations,aconsistentperturbationtheorycanbeapplied togetapproximatevaluesoftheenergylevelsandofthesecondconstantofmotion.

©2016ElsevierB.V.All rights reserved.

1. Introduction

Ina previous paper [1]we have considered a kindofLandau systemwith two charged particles in a plane. The charges have the same absolute value and opposite sign and the same mass. Theseparticles are subjectto a constant perpendicular magnetic field.Thesamesituation,underadifferentpointofviewhasbeen discussedinsomerecentpapers[2,3].

This model has considerable interest in Physics. For instance, itcan beinterpreted asa positroniumsystem, orasaFrenkel or Mott–Wannierexciton[4].Otherapplicationshavebeenstudiedin

[5,6].Themodelisalsocloselyrelatedtothesystemofaparticle undertwofixedgravitycenters,aclassicalsubject[7].

We haveshown in [1]that this system maybe studied from the point of view of either classical or quantum mechanics. The transitionfromtheformertothelatterisachievedthrough canon-icalquantization[8].Thissystemhasfourindependentcommuting constantsofmotion,orsymmetries.Classically,thecommutationis definedintermsofPoissonbrackets.Thissystemisintegrable al-thoughnotsuperintegrable.

Intheclassicalanalysispresentedin[1],we haveusedtwoof theseconstants,writtenincompactformasthecomponentsofthe

*

Correspondingauthor.

E-mailaddresses:[email protected](J.S. Ardenghi),

[email protected](M. Gadella),[email protected](J. Negro).

twodimensionalvector

μ

,inordertoreduce bytwothenumber of degreesof freedom, so that we have an effective two dimen-sionalsystem.

TheresultingHamiltonianisasumofakinetictermplusan ef-fectivepotential,whichisgivenbythesumofaCoulombpotential plusashiftedharmonicoscillator.Alongwiththiseffective Hamil-tonian, we havean additional constant ofmotion,denoted by T. Thisfactallowstoseparatethesysteminellipticcoordinates.

In the present Letter, we focus our interest on the quantum version of thismodel. Within thisquantum context, the separa-tion inellipticcoordinates oftheeffectivesystemleads to a pair ofequations.Oneis afourthdegree periodicHillequation,while thesecondoneisasimilarmodifiedHillequationwithhyperbolic functions[9,10].Uptoourknowledge,analytic solutionsforthese equationsarenotknown.

Alongthispresentation, we shalldiscussthe possibilityof ob-taining approximate solutions of these equations by means of a procedure based on perturbation theory. In our calculations, we shalluse

μ

:= |

μ

|

asnaturalperturbativeparameter.This

μ

,tobe definedin thenext section (right after(6)), isa constant of mo-tionandgivesthepositionofthecenterofthedisplacedharmonic oscillator.

Thezeroorderofperturbationwillapproximatelydescribethe system either for

μ

<<

1 or for

μ

>>

1. In the first case, the Coulomb term will be dominant with respect to the oscillatory term,nowusedasaperturbation.Thesituationisreversedinthe

http://dx.doi.org/10.1016/j.physleta.2016.03.038

1818 J.S. Ardenghi et al. / Physics Letters A 380 (2016) 1817–1823 secondcase, wheretheoscillatorytermisdominantandthe

per-turbationisgivenbytheCoulombpart.

We shall see that in the zero order approximation valid for

μ

<<

1, or Coulomb approximation, both trigonometric and hy-perbolic Hill equations become a pair of equations of the type discussed by Razavy in [11,12], which are solvable. These equa-tionswereknownbysomeauthorsasthehyperbolicWittaker–Hill equations. Nevertheless,since we are using asthe referencethe work by Razavy and following the use of some recent authors, weprefertousetheterminologyRazavytypeequationsorsimply,

Razavy equations. Their solutions coincide withthe bound solu-tionsoftheCoulombproblemandtake significantvaluescloseto the originof the potential. This means that the relative position betweenbothparticleskeepsverysmall.

When

μ

>>

1, wecan againapproximate bothHill equations by Razavy type equations. Their solutions describe the solutions fortheharmonicoscillatorinellipticcoordinates.Thesesolutions aretheboundstatesoftheharmonicoscillatorandcorrespondto muchlargervaluesof

μ

.

Usingthesezeroorderapproximationsastheunperturbed sys-tems,weproposeperturbationsoffirstorder.Inthisapproach,we assume that therepresentationin termsofelliptic coordinatesis validforanyorderofperturbation.

This Letter is organized as follows: In Section 2, in order to orient the reader andfor the sake of completeness, we summa-rizetheresultsobtainedin[1]withsome additionalinformation. WegiveinSection3theexactresolutionofthezeroorderRazavy equations,wherewepayanspecialattentiontothecorrectchoice oftheboundaryconditionsanditsconsequences.InSection 4,we discussthefirstorderperturbative approachto solutions.Explicit expressionsare leftto SupplementaryMaterial.We closeour dis-cussionwithsomeConcludingRemarks.

2. Presentationoftheproblem

Inthissection,webrieflyreviewthetreatmentgivenin[1].Let usbegin withthe classical descriptionof the model.The Hamil-tonian describing two charged particles, with charges e and

−

e, of equal massm, interacting amongthemselves by the Coulomb potential andsubject to an external constant magnetic field per-pendicular to the plane in which the particles move is given by (c

=

1):

H

=

1

2m

[

(

p

(1)

₋

_{e A}

₍

_x(1)

₎₎

2

₊

₍

_p(2)

₊

_{e A}

₍

_x(2)

₎₎

2

_{] −}

e2

|

x(1)

₋

_x(2)

_|

.

(1)

Here by x(k), p(k), k

=

1

,

2, we denote positions and linear mo-mentaofboth particles.The vectorpotential A

(

x

)

istakeninthe symmetricgauge,

Ai

(

x

)

=

h

ε

i jxj

,

A

(

x

)

=

(

h x2

,

−

h x1

) ,

(2) where

ε

i j isthe totally antisymmetric tensorin two dimensions. Weareusingtheconventionofsummationoverrepeatedindices. ThemagneticfieldisparalleltothezaxiswithintensityB

= −

2h. Foreach particlek (k

=

1

,

2), we define a kinematicmomentum

π

(k)_{withcomponents,}

π

_i(1)

=

p_i(1)

−

e A(_i1)

=

p(_i1)

−

e

ε

i jx(_j1)h

,

π

_i(2)

=

p_i(2)

+

e A(_i2)

=

p(_i2)

+

e

ε

i jx(_j2)h

.

(3) Next,definethecomponents, iofthetotalmomentumandthe centerofmass(c.o.m.)coordinates Qi,i

=

1

,

2,by

i

:=

π

i(1)

+

2eh

ε

i jx(j1)

+

π

(2)

i

−

2eh

ε

i jx(j2)

,

Qi

:=

1 2

(

x

(1)

i

+

x

(2)

i

)

(4)

andtherelativemomentumandcoordinatesas

π

i

:=

1 2

(

π

(1)

i

−

π

(2)

i

) ,

qi

:=

x

(1)

i

−

x

(2)

i

,

i

=

1

,

2

.

(5) Due to thefact that the totalcharge ofthe systemvanishes, the functions

{

q

,

π

,

Q

,

}

constitute a canonical coordinate set. We could have equally discussed the case of two particles with dif-ferentmassm1,m2;then,asimilarcanonicalsetwouldhavebeen obtained.

In termsof thesenewcoordinates, theinitial Hamiltonian(1)

hasthefollowingform:

H

=

1

4m

2

₋

eh

m

ε

i j

iqj

+

1

m

π

2

₊

e2h2

m q

2

₋

e2

q

=

1

m

π

2

₊

e2h2

m

(

q

+

μ

2

)

2

₋

e2

q

,

(6)

withq

:= |

q

|

,

μ

j

= −

ε

i j

i

/

ehand

μ

:= |

μ

|

.AsthecoordinatesQ are cyclic, the components ofthe “total momentum”

are con-stantsofmotiongivenintermsof

μ

.From (6),we concludethat the effective system consists of a particle, with a reduced mass

m

/

2 andcharge e,intheplaneundertheinfluenceofaCoulomb potentialsetattheoriginwithcharge

−

e,plusashiftedharmonic oscillator potential with angular frequency

ω

=

2eh

/

m

=

e

|

B

|

/

m, whichisthecyclotron frequency,being

|

B

|

=

2hthemagneticfield intensity.

As is clear from(4),the constant of motion

is the sumof the generators of magnetic translations for each particle,just as they are defined forthe Landausystem of a single particle in a constantmagneticfield.IntheLandausystem,thevaluesof

give thecenterofthecirculartrajectories;inthiscase,thevaluesof

determinetherelativepositionofthetwocentersoftheCoulomb andoscillatoreffectivepotentials.

Asthefirstterm, _4m1

2

=

(eh_4mμ)2,in(6)isaconstant,itwillbe hereafterdroppedtosimplifytheexpressions.Nevertheless,itwill berecoveredlaterinordertointerprettheapproximation

μ

>>

1. Asshownin[1],thissystemhastwoindependentconstantsof motion:

H

=

π

2

m

+

U

(

q

)

;

T

:=

π

igi j

(

q

)

π

j

+

(

q

) .

(7)

Here, H is the effective Hamiltonian given in (6), without the abovementionedconstantterm.ThesecondconstantofmotionT

includesa“kineticterm”givenby1_:

π

igi j

(

q

)

π

j

=

L2

+

(

μ

1

π

2

−

μ

2

π

1

)

L

=

1 2

(

L

·

L

₊

_L

_·

_L

_{) ,}

L

:=

q1

π

2

−

q2

π

1

,

(8)

where LandLaretheangularmomentawithrespecttothe ori-gin andto thepoint

−

μ

,respectively. Bythe way,thistermhas been alreadyobtainedby Erikson–Hill in[14] for thetwo center problem.Thekinetictensorcanalsobeexpressedas

gi j

=

q∗iq∗j

+

1 2

(

μ

∗

iq∗j

+

q∗i

μ

∗j

) ,

(9) where,q∗_i

=

ε

ikqk,

μ

_i∗

=

ε

ik

μ

k.The“potentialterm” isgivenby

(

q

)

=

2m e

2

μ

2 q

·

μ

q

+

e2h2

4

(

q

2

_μ

2

₋

₍

_q

_·

_μ

₎

2

_{) .}

₍₁₀₎

NotethatT isaconstantofmotioninthesensethat

{

H

,

T

}

=

0, where

{·

,

·}

stands forPoisson bracket. Then with thehelp of T, wecanseparatethesystemusingtheconfocalellipticcoordinates

(

α

,

β)

,wherethe fociare setattheoriginandat

−

μ

.Theseare definedby[1]:

μ

cosh

α

=

q

+ |

q

+

μ

|

,

−

μ

cos

β

=

q

− |

q

+

μ

|

.

(11)

Forinstance,intheparticularcase

μ

=

(

μ

,

0

)

,wehavethe follow-ingexpressionoftheCartesiancoordinatesintermsoftheelliptic coordinates:

x

+

μ

2

=

μ

2 cosh

α

cos

β,

y

=

μ

2 sinh

α

sin

β .

(12)

According to (12), taking either the ranges

α

∈ [

0

,

+∞

)

,

β

∈

[−

π

,

π

)

or

α

∈

(

−∞

,

+∞

)

,

β

∈ [

0

,

π

)

, we cover the whole real plane.

Oncewehaveintroducedthemodelclassically,wecanproceed withitsquantumversion.Inthiscontext,wereplacethefunctions

H andT in(8)bytheoperators:

H

:= −

1

m

∂

i

∂

i

+

U

(

q

)

;

T

:= −

∂

igi j

(

q

)∂

j

+

(

q

) ,

∂

i

≡

∂

∂

qi

,

(13)

respectively (h

¯

=

1) following the canonical quantization proce-dure [8]. In orderto avoid unnecessary notational complications, wehaveuseHandT forbothclassicalfunctionsandquantum op-erators.Then,theeigenfunctionequationsfortheseoperatorsread, respectively:

(

H

−

E

)(

x

)

=

0

,

1

mT

−

J

(

x

)

=

0

,

(14)

wheretheparametersE and

J

givethecorrespondingeigenvalues ofthe HamiltonianH andthesymmetry operator T.These equa-tions canbe separatedin termsof theelliptic coordinates

(

α

,

β)

withtheusualAnsatzthatthesolution

(

α

,

β)

befactorizablein termsofthesecoordinates [1]:

(

α

,

β)

=

ψ(

α

)φ (β)

.Thus, equa-tions(15)thatwerecoupledinCartesiancoordinates,giverisetoa systemoftwoordinarysecondorderdifferentialequationsin sep-aratedvariables,forwhichtheirexplicitformisgivenby

d2

ψ (

α

)

d

α

2

=

m

Asinh4

α

−

Bcosh

α

−

Csinh2

α

+

J

ψ (

α

) ,

(15)

−

d2

φ (β)

d

β

2

=

m

Asin4

β

+

Bcos

β

+

Csin2

β

+

J

φ (β) ,

(16)

where

A

=

e

2_h2

m

_μ

2

4

,

B

=

e2

μ

2

,

C

=

E

μ

2

4

.

(17)

Firstofall,note thatalthoughequations(15)and(16)are sep-arated in the variables

α

and

β

, they are not separated in the variables E (throughC) and

J

,whicharetheconstants ofmotion thatwewishtoobtain.

Next,theseequationsshouldsatisfysomeboundaryconditions and, hence, we are facing to a sort of Sturm–Liouville problem. Since

β

isanangle,thesolutions

φ (β)

of(16),whichweare look-ing for, must be periodic withperiod 2

π

. Equation (15) can be consideredasaSchrödingerequationforthefunction

ψ(

α

)

onthe wholerealline

−∞

<

α

<

∞

.Thus,thesolutions

ψ(

α

)

of(15) de-scribingboundstatesshouldsatisfythat

ψ(

α

)

→

0,sufficientlyfast as

α

→ ±∞

.Inaddition,asthefunctionsbetweenbracketsin(15)

and(16)are even,thentheir respectivesolutions

ψ(

α

)

and

φ (β)

canbechosentohaveawell definedparity.Aswe mayconclude afterthecommentfollowing(11),therangegivenby

α

∈

(

−∞

,

∞

)

and

β

∈

(

−

π

,

π

)

isovercomplete.Thus, inorderthatthesolution

=

ψ(

α

)φ (β)

beasinglevaluedfunctionon

R

2_{,we mustselect} thesolutions

ψ(

α

)

and

φ (β)

havingthesameparity.

Note that equations (15) and (16) are indeed quite similar, which willhelp usto obtainsolutions forthis system. Itis clear that (15) can be obtained from (16) by the replacement

α

→

i

(β

+

π

)

.Now,assumethatwehavefoundasolution

ψ(

α

)

of(15)

withwell definedparity, analytic on

α

andsatisfyingthe proper boundaryconditions.Then,thereplacement

α

→

i

(β

+

π

)

willgive us a solution of (17) with the same parity and, hence, a single valuedsolution

(

α

,

β)

.Consequently,beinggivenananalytic so-lutionof(15)withthe properboundaryconditions,thisprovides uswithacompletesolutiontotheproblem.

Finally,weneedtoanalyzetheformofequations(15)and(16). Tothisend,letusconsiderthefollowingtypeofsecondorder dif-ferentialequations:

d2

ψ (

x

)

dx2

+

A0

+

2

∞

n=1

Ancos

(

2nx

)

ψ (

x

)

=

0

,

(18)

whereA0

,

A1

,

A2

,

. . .

areconstants.Thisisan-thdegreeHill equa-tion if An

=

0 and An+1

=

An+2

= · · · =

0. The first degree Hill equation is called the Mathieu equation. The second degree Hill equation hasbeennamedtheWhittaker–Hillequation. Upto our knowledge, no systematicstudies of Hillequations of higher de-greehasbeendone.Inparticular(16)isafourthdegreeHill equa-tionforwhichthesolutionsarenotknown.

In thepresentpaper, we attemptto obtainapproximate solu-tionsfor E and

J

.Thiswillbedone inthenexttwosectionsand intheSupplementaryMaterial.

3. Twosolvableapproximations

In this section, we shall discuss two approximations to the problemunderdiscussionthathaveexactsolution.These approxi-mationsareappropriateforsmallandhighvaluesoftheparameter

μ

asdefinedintheprevious section.Bothapproximationsreduce the fourth degree Hill equations into the solvable Razavy equa-tion.Thefirstapproximationisvalidfor

μ

<<

1 andiscalledthe Coulomb limitasgivesan energyspectrum ofCoulomb type.The other is valid for

μ

>>

1 and is named the harmonic oscillator limit as its energy spectrum is the corresponding to an off cen-teredharmonicoscillator.

Observethat the Coulomb limitis justthespecial caseh

=

0, whichisequivalenttoA

=

0.Ontheotherhand,theharmonic os-cillatorlimit,B

=

0,canbetakene

→

0 withh

→ ∞

iftheproduct

ehremainsconstant.Inaddition,since A

=

O

(

μ

4

₎

_,B

₌

_O

₍

_μ

₎

_and

C

=

O

(

μ

2

₎

_{, the large andsmall}

_μ

_{limits correspond to the} har-monicoscillatorandtheCoulomblimit,respectively.

3.1. TheCoulombapproximation

μ

<<

1

Ifweconsider

μ

small,thetermwithcoefficient A in(15)and

(16)isnegligible,duetotheformofthecoefficientsgivenin(17). Inourzerothorderapproximation,A

=

0.Then,equations(15)and

(16)takethefollowingform:

d2

ψ (

α

)

d

α

2

= −

m

Bcosh

α

+

Csinh2

α

−

J

ψ (

α

) ,

(19)

d2

_{φ (β)}

d

β

2

= −

m

Bcos

β

+

Csin2

β

+

J

φ (β) .

(20)

The Schrödinger equation (19) has bound solutions with right boundary conditions in

α

→ ±∞

, provided that C

<

0. Accord-ingto(17),thismeansthattheenergyisnegative,i.e., E

<

0.This equationhasexactsolutionsasitcanbetransformedintoaRazavy equation,alsocalledthehyperbolicWittaker–Hillequation, which hastheform[11,13,15,16]

−

d2

ψ (

x

)

dx2

+

(ζ

cosh 2x

−

M

)

1820 J.S. Ardenghi et al. / Physics Letters A 380 (2016) 1817–1823 Table 1

Valuesoftheconstantsofmotionandtheeigenfunctionsforn=0,1,2 andμ<<1. E0= −me4 λ01=1+μ2m2e4 J01=0

ζ0=μme2 ψ01(α)=exp(−ζ20coshα)

E1= −me 4

4 J11=1+μme

2 4m ζ1=μme

2

2 ψ11(α)=cosh(α/2)exp(− ζ1

2coshα)

J12=1−μme 2 4m

ψ11(α)=sinh(α/2)exp(−ζ21coshα)

E2= −me 4

9 J21=m1

ζ2=μme 2

3 ψ21(α)=sinhαexp(−ζ22coshα)

J22= 1+1+4

9μ2m2e4 2m

ψ22(α)=(coshα− 1−1+4ζ2

2 ζ2 )exp(−

ζ2 2coshα)

J23= 1−1+4

9μ2m2e4 2m

ψ23(α)=(coshα− 1+1+4ζ2

2 ζ2 )exp(−

ζ2 2coshα)

Thistransformation goesasfollows:first performthat change ofvariables

α

=

2xand,then,somestraightforwardmanipulations toobtainfrom(19)thefollowingrelation:

−

ψ

(

x

)

+

_√

−

4mCcosh 2x

−

√

2mB

−

4mC

2

ψ (

x

)

= −

mB2

C

+

4mC

+

4m

J

ψ (

x

) .

(22)

Equation(22)hasalreadytheform(21)withthefollowing identi-fications:

ζ

=

√

−

4mC

=

μ

√

−

m E

,

(23)

M

=

√

2mB

−

4mC

=

e

2

−

m

E

,

(24)

−

λ

=

mB

2

C

+

4mC

+

4m

J

.

(25)

TheRazavyequation(21)issolvableanditssolutions

ψ(

x

)

are squareintegrable, providedthat M beapositiveinteger, i.e., M

=

n

+

1,withn

=

0

,

1

,

2

,

. . .

.

Letusdenote the energylevels by En(C),n

=

0

,

1

,

2

. . .

, where thesuperscript

(

C

)

standforCoulomb.Then,ifM2

=

(

n

+

1

)

2,we have

En(C)

= −

me4

(

n

+

1

)

2

.

(26)

Needlesstosaythat thesimilarityofthe energylevels in(26)

with the energy levels for the Coulomb problemis obvious. We recall that ourHamiltonian(6) hasa potential termof harmonic oscillatortypeandotherofCoulombtype.Weconcludethatwhen

μ

<<

1,theCoulombtermpredominatesovertheharmonic oscil-lator.However,notallenergylevels(23)maybephysically admis-sible,aswe havenottakenintoaccounttheboundary conditions yet.

Beingfixed E(_Cn),thevalue of

ζ

is obtainedwith(23).We de-note it by

ζ

n. Then, we replace M

=

n

+

1 and

ζ

n in the Razavy equation(21),soastoobtainthen

+

1 knownsolutionsfor

λ

.We denotethesesolutions as

λ

n,1

,

. . . ,

λ

n,n+1.Each

λ

n,r determines a value

J

n,rthrough(25).

Insummary,each energylevel E(_Cn) hasadegeneracyof order

n

+

1,whichischaracterizedbythen

+

1 valuesof

J

n,r.Wedenote

theircorrespondingsolutionsas

ψ

n,r

(

x

)

.

InTable 1,we give thevalues of En,

ζ

n,

J

n,r and

ψ

n,r

(

α

)

,the latterinterms oftheoriginal variable

α

=

2x,forn

=

0

,

1

,

2. For

highervaluesofn,thesepolynomialsolutionsareofhigherdegree and,therefore,theybecamemoreandmorecomplicated.

Forequation(20),wejustneedtoreplace

α

byi

(β

+

π

)

.Then,

(20)becomes(19)whichisalreadysolved.Theeigenfunctionsare oftheform

φ

n,r

(β)

withthesamevaluesoftheenergyand

J

n,r.

However, thisisnotthewhole storybecausesolutionsof(20)

mustfulfillperiodicboundaryconditionsofperiod2

π

.Inaddition, they should be ofthe sameparity asthesolutions of(19).From

Table 1, and after the transformation

α

→

i

(β

+

π

)

, it is rather trivialtoprovethattheseconditionsaresatisfiedforevenvaluesof

nonly.Thisshowsthatthecorrectsolutionscorrespond toenergy levelsoftheform

E(_jC)

= −

me

4

(

2j

+

1

)

2

,

j

=

0

,

1

,

2

, . . .

(27)

whichpreciselycoincidewiththeenergylevelsoftheCoulomb po-tentialontheplane.Thegroundstate solution 0,1

(

α

,

β)

isgiven by

01

=

ψ

01

(

α

)φ

01

(β)

=

N e−

ζ₀

2(coshα−cosβ)

=

_{N e}−me2q

_.

₍₂₈₎

As a matter offact,one can realizethat thesystem (19)–(20)

gives the two dimensional Coulomb problem in elliptic coordi-nates, see [17]. We have shown that the exact solutions to this problemcanbeobtainedfromthepropertiesoftheRazavy equa-tion.SeeFigs. 1 and2.

3.2. Theharmonicoscillatorapproximation

μ

>>

1

The second approximation has also exact solutions and cor-responds to the case

μ

>>

1, where the term containing B is dropped out after (17). The consequence is that the system be-haves nowasan out ofcenterharmonicoscillator.For complete-ness,wewritetheequations(16)and(16)withoutthetermonB:

d2

ψ (

α

)

d

α

2

=

m

Asinh4

α

−

Csinh2

α

+

J

ψ (

α

) ,

(29)

d2

_{φ (β)}

d

β

2

= −

m

Asin4

β

+

Csin2

β

+

J

φ (β) .

(30)

Aftersomestraightforwardtransformations,(29)comesinto

−

d2

ψ (

α

)

d

α

2

+

m A

4 cosh

(

2

α

)

−

1 2

m A

(

A

+

C

)

2

−

m

4A

(

A

+

C

)

2

ψ (

α

)

=

m

A 4

+

C

2

+

J

ψ (

α

) .

(31)

Theresultingequation (31)isagainaRazavyequationofsolvable typewithparametersgivenby

ζ

=

m A

4

,

M

=

1 2

m

A

(

A

+

C

)

=

n

+

1

,

λ

=

m

4A

(

A

+

C

)

2

₊

_m

A

4

+

C

2

+

J

(HO)

.

(32)

Thesuperscript

(

HO

)

in(32)meansharmonicoscillator.Weknow that(31)issolvablewithsquareintegrablesolutionsprovidedthat

M

=

n

+

1 withn

=

0

,

1

,

2

,

. . .

. Then, we recover A and C from

(17)anduseitin thesecond equationofthefirst rowin(32)to concludethat

E(nHO)

=

2eh

m

(

n

+

1

)

−

(

eh

μ

)

2

Fig. 1.Graphics of the ground eigenfunctionψ01(left) and the first excited eigenfunctionψ21(right) inTable 1.

Fig. 2.Form of the potential.

Table 2

Valuesoftheconstantsofmotionandtheeigenfunctions,uptonormalization,for n=0,1,2 andμ>>1 (theharmonicoscillatorlimit).Thewavefunctionsdepend ontheconstant˘e=eμ2_throughζ.

E∗0= 2eh

m J01=−64ζ+ehμ 2_(−16+ehμ2₎ 64m ζ=eh8μ2 ψ01(α)=exp(−ζ20cosh 2α)

E∗1= 4eh

m J11=−64+64(−2+ζ )+ehμ 2₍₋32+ehμ2₎ 64m

ζ=ehμ2

8 ψ11(α)=cosh(α)exp(−ζ2cosh 2α)

J12=−64+64(2+ζ )+ehμ 2_(−32+ehμ2₎ 64m

ψ11(α)=sinh(α)exp(−ζ2cosh 2α)

E∗₂=6eh

m J21=−2304+eμ

2(64e3m2₊9h(−48+ehμ2)) 576m

ζ=ehμ2

8 ψ21(α)=sinh 2αexp(−

ζ 2cosh 2α)

J22=eμ

2_(64e3_m2_{+9h(−48+ehμ}2_{))−384(3+9+4e}4m2μ2) 576m

ψ22(α)=(cosh 2α−1−

1+4ζ2 ζ )exp(−

ζ 2cosh 2α)

J23=eμ

2_(64e3_m2_{+9h(−48+ehμ}2_{))−384(−3+9+4e}4m2_μ2) 576m

ψ23(α)=(cosh 2α−1+

1+4ζ2 ζ )exp(−

ζ 2cosh 2α)

which corresponds to the energylevels of a displaced harmonic oscillator.Notethat theshiftofenergyin(33)coincides withthe droppedtermintheHamiltonian(6).Inourcommentsafter equa-tion(6),wehaveannounced thatthat termshouldreappear. The valuesforthesecondconstantofmotion

J

(HO)andthewave func-tionsareobtainedasusual[11,13,15,16,22].InTable 2,welistour resultsforthethreefirstenergylevels.

Fromthesolutionsof(29),weobtainthesolutionsof(30)just withthereplacement

α

→

i

(β

+

π

)

,asintheCoulombcase. How-ever,thereisanimportantdifferencebetweenbothsituations.We seein Table 1that the functions

ψ

nr

(

α

)

forCoulomb andn odd dependon

α

/

2 whichimpliesthatthefunctions

φ

nr

(β)

,fornodd, donot haveperiodicitywithperiod2

π

and, therefore,we hadto discardthesesolutions.Thisisnotthecasenowwiththeharmonic oscillatorapproximation,where alleigenfunctions

ψ

nr

(

α

)

depend on

α

or2

α

. Then, thefunctions

φ

nr

(β)

obtainedby thismethod are periodic withperiod 2

π

even for n odd.In consequence, all

energy valuesshould be considered, even those withn odd.The groundstateisnow

(

α

, β)

=

exp

(

−

ζ

0

2

(

cosh 2

α

+

cos 2

β))

∝

exp

(

−

eh

2

(

q

+

μ

/

2

)

2

_{) .}

₍₃₄₎

Notethatthisisthegroundstateofaharmonicoscillatorcentered at

−

μ

/

2.Infact,wehaveobtainedthesolutionsfortheharmonic oscillatorseparatedinellipticcoordinates[18,19].

Sofar,we haveproposedtwoexactlysolvableapproximations. What happensif

μ

= |

μ

|

is neithertoo smallnortoo big?Then, we proposea perturbative treatmenttakingthe previous approx-imations asthe unperturbedsolution.Astheperturbed potential, weusethedroppedtermforeachcase.Thiswillbetheobjective ofthenextsection.ResultsaregivenintheSupplementary Mate-rial.

4. Aperturbativeapproximationtothecompletesolution

In the sequel, we are dealing with the quantum case only. Therefore,expressions likeH,H0,H1,T,etc.,willhenceforth rep-resent operators.Wefollowthisconventioninordertousea no-tationassimpleaspossible.

In this section, we are looking for approximate solutions for equations(15)and(16).Letusbeginwiththeunperturbed equa-tions:

(

H0

−

E0

)

0

(

x

)

=

0

,

1

m T0

−

J

0

0

(

x

)

=

0

,

(35)

where H0 andT0 representunperturbedversions of H and T in

(13).Theyareassumedtobevalidfortheapproximations

μ

<<

1 and

μ

>>

1.Therefore,equations(35)leadtoequations(19)and

(20)for

μ

<<

1 and (29)and(30)for

μ

>>

1. Inthe firstcase, thetermcontaining A inequations(15)and(16)isomitted,while thetermcontaining Bdropsoutinthesecond.

Weintendtogivea perturbativefirstorderapproximation,for whichwejustwrite

H

=

H0

+

H1

,

T

=

T0

+

T1

,

(36)

E

=

E0

+

E1

,

J

=

J

0

+

J

1

,

(37)

=

0

+

1

.

(38)

Note that H0 and T0 commute and also does H and T. This isa crucialpoint here, asimpliesthat thefunction

showedin

(38)is asimultaneous eigenfunctionof H andT.Thus, replacing

1822 J.S. Ardenghi et al. / Physics Letters A 380 (2016) 1817–1823

(

H0

+

H1

)(

0

+

1

)

=

(

E0

+

E1

)(

0

+

1

) ,

(39)

1

m

(

T0

+

T1

) (

0

+

1

)

=

(

J

0

+

J

1

)(

0

+

1

) .

(40)

Equations(39)and(40)yieldthedesiredresultusingthestandard procedureinperturbationtheory[20,21]:

E1

=

0

|

H1

0

0

|

0

,

J

1

=

0

|

T1

0

0

|

0

.

(41)

TofindH1andT1,wefirstwriteU

(

q

)

and

(

q

)

in(7)interms of elliptic coordinates. The explicit forms of U

(

q

)

and

(

q

)

are takenfrom[1]andtheirexpressionintermsofellipticcoordinates istheresultofastraightforwardcalculation:

U

(

q

)

= −

e

2

q

+

e2h2 m

(

q

2

₊

_q

_·

_μ

₎

= −

2e2

μ

1

cosh

α

−

cos

β

+

e2h2

μ

2

4m

(

sinh

2

_α

₋

_sin2

_{β) ,}

₍₄₂₎

(

q

)

=

2m e

2

μ

2 q

·

μ

q

+

e2h2

4

(

q

2

_μ

2

₋

₍

_q

_·

_μ

₎

2

₎

= −

me2

μ

2

cosh

α

cos

β

−

1 cosh

α

−

cos

β

+

e2_h2

_μ

4

16 sinh

2

_α

_sin2

_{β .}

₍₄₃₎

For completeness, we here supply the expression of the “kinetic terms”ofbothconstantsofmotion:

1

m

π

2

₌

4

m

μ

2

1 sinh2

α

+

sin2

β

∂

2

∂

α

2

+

∂

2

∂β

2

(44)

π

igi j

π

j

=

1 sinh2

α

+

sin2

β

−

sin2

β

∂

2

∂

α

2

+

sinh 2

_α

∂

2

∂β

2

.

(45)

In order to assign an explicit form to the terms in the split

(36)–(37),wegobacktoequations(7)–(8)towritethemas

H

=

π

2

m

+

U

(

q

)

=

π

2

m

+

U0

(

q

)

+

U1

(

q

)

=

H0

+

U1

(

q

)

(46)

and

T

=

π

igi j

(

q

)

π

j

+

(

q

)

=

π

igi j

(

q

)

π

j

+

0

(

q

)

+

1

(

q

)

=

T0

+

1

(

q

) ,

(47)

whereU1

(

q

)

=

H1 and 1

(

q

)

=

T1.

We observefrom(44) that thekinetic termin H dependson

μ

−2_{, so that it may be convenient redefining H by multiplying}

(46)by

μ

2_{.Thefinalresultis}

H

=

μ

2

π

2

m

+

μ

2_U

₍

_q

₎

₌

_μ

2

π

2

m

+

μ

2_U

0

(

q

)

+

μ

2U1

(

q

)

=

μ

2

π

2

m

+

U0

(

q

)

+

U1

(

q

)

=

H0

+

U1

(

q

) ,

(48)

wheretheformof

H0 isobvious.Inthe SupplementaryMaterial, wegivetheexplicitfirstordercorrectionsinbothcases.

5. Concludingremarks

Integrabilityandsolvabilityarequitedifferentconcepts.Thisis illustrated inthe presentletter, wherewe havediscussed an ap-parentlyvery simplemodel: theLandauproblemresultingofthe considerationoftwochargedparticlesofoppositesignonaplane subjecttoaperpendicularconstantmagneticfield.

Asshowninapreviouspaper,thissystemisequivalenttoone inwhichaparticleissubjecttoaCoulombpotentialplusashifted harmonicoscillator. Inthesearch for solutionsherelabelled (15)

and(16),we haveusedseparationofvariablesintermsofelliptic

coordinates,aprocedurewhichyieldstotwoseparatedequations. Althoughtheseequationsare notsolvable, wehaveanalyzedtwo solvableapproximationsdependingonthevaluesofa characteris-ticparameter

μ

.

For small values of the parameter

μ

, i.e.,

μ

<<

1, we omit thehighestordertermon

μ

intheequations.Thistermcontains the coefficient A,which dependson

μ

. Theresulting situationis a Coulomb system expressed inelliptic coordinates described by two equations separatedon the coordinates.These equations are exactlysolvableofRazavytypeandtheirsolutionsgivethebound states for the Coulomb system. We justify the approximation by noting,makinguseoftheapproximatesolutions,thatthedropped term,i.e.,

(

Asinh4

α

)

ψ(

α

)

,isverysmallforanyrealvalueof

α

.

For large values of

μ

,

μ

>>

1, the fundamental equations

(15)–(16) can be simplified if we drop the term with the coef-ficient B. The resulting approximation is now equivalent to dis-placedharmonicoscillatorinellipticcoordinates.Afterthis simpli-fication, equations (15)–(16) areagain solvableRazavy equations. InTable 2,we givethe wavefunctionscorrespondingto the low-estenergylevels.Thesesolutionsspreadout acenterofpotential located at

−

μ

/

2. Again,an useofthese solutions show that the droppedterm,inthiscase

(

Bcosh

α

)ψ(

α

)

isverysmallforallreal valuesof

α

.

Equations(15)–(16)aswellasthoseresultingofthementioned approximationscanbeobtainedfromeachotherjustbeusingthe correspondence

α

→

i

(β

+

π

)

.Consequently,oncewe havefound asolutionfortheequationon

α

,wehaveacorrespondentsolution forthe equationon

β

.However, not allthesesolutions are phys-ical,onlythosefulfillingthecorrectboundaryconditionsforboth variables.

Taking these solvableapproximations asa point of departure, wehavestudiedfirstorderperturbativecorrections.These pertur-bationsshowabreakofthedegeneracyoftheinitialenergylevels. It is noteworthy that both zero order systems(Coulomb and os-cillator)andthetotalsystemincludingtheperturbationtermsare separatedwithrespecttothesameellipticcoordinates.

The approximations

μ

<<

1 and

μ

>>

1 shouldbe compared withthebehaviorofthesystemonthelimits

μ

→

0 and

μ

→ ∞

, respectively. In the limit

μ

→

0, both potentials, Coulomb and harmonic oscillator remain. Both centers are located at the ori-gin. In this limit, T

∼

L2 and elliptic coordinates become polar coordinates [3]. This is consistent with the rotational geometric symmetryaroundthe z-axisofthesystematthislimit.Theother situation comes inthe limit

μ

→ ∞

.Here, after a translationof coordinates,thewholesystembecomesaharmonicoscillator cen-teredattheorigin,withHamiltonian

H

=

π

2

m

+

e2h2

m q

2

_.

Then, taking

μ

:=

(

μ

,

0

)

, the second constant of motion T takes theform:

T

∼

π

2 y

m

+

e2h2

m q

2 y

,

so that T appears in termsof Cartesian coordinates viewedas a limitofellipticcoordinates.

Acknowledgements

We acknowledge partial financial support to the Spanish MINECO(ProjectMTM2014-57129-C2-1-P).

Theauthorsalsowishtoacknowledgetheexcellentworkofthe reviewersthathelpedtoimprovethepresentmanuscript substan-tially.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttp://dx.doi.org/10.1016/j.physleta.2016.03.038.

References

[1]M.Gadella,J.Negro,L.M.Nieto,G.P.Pronko,Twochargedparticlesintheplane underaconstantperpendicularmagneticfield,Int.J.Theor.Phys.50(2011) 2019–2028.

[2]M.A.Escobar-Ruiz,A.V.Turbiner,TwochargesonplaneinamagneticfieldI. “Quasi-equal”charges andneutralquantumsystematrestcases,Ann.Phys. 340(2014)37–59.

[3]M.A.Escobar-Ruiz,A.V.Turbiner,Twochargesonaplaneinamagneticfield:II. Movingneutralquantumsystemacrossamagneticfield,Ann.Phys.359(2015) 405–418.

[4]L.P.Gorkov,I.E.Dzyaloshinskii,ContributiontothetheoryoftheMottexciton inastrongmagneticfield,Sov.Phys.JETP26(1968)449–451.

[5]N.R.Kestner,O.Sinano˘glu,Studyofelectroncorrelationinhelium-likesystems usinganexactlysolublemodel,Phys.Rev.128(1962)2687.

[6]N.H. March, J. Negro, L.M. Nieto, Kinetic energy and ground-state elec-tron densities as fingerprints of wavefunction entanglement in two-electronspin-compensatedatomic models,J.Phys. A,Math.Gen.39(2006) 3741–3751.

[7]W. Miller Jr., A.V. Turbiner, Particle in a field of two centers in prolate

spheroidalcoordinates:integrabilityandsolvability,J.Phys.A,Math.Theor.47 (2014)192002.

[8]M. Gadella,J.M. GraciaBondia,L.M.Nieto,J.C. Varilly,Quadratic Hamiltoni-ans in phasespace quantummechanics, J. Phys. A, Math. Gen. 22(1989) 2709–2738.

[9]W.Magnus,S.Winkler,Hill’sEquation,Dover,NewYork,1979.

[10]G.Teschl,OrdinaryDifferentialEquationsandDynamicalSystem,AMS, Provi-dence,RI,2012.

[11]M.Razavy,AnexactlysolubleSchrödingerequationwithabistablepotential, Am.J.Phys.48(1980)285–288.

[12]M.Razavy,Apotentialmodelfortorsionalvibrationsofmolecules,Phys.Lett. A82(1981)7–9.

[13]H.Konwent,One-dimensionalSchrödingerequationwithanewtype double-wellpotential,J.Phys.A118(1986)467–470.

[14]H.A.Erikson,E.L.Hill,Anoteontheone-electronstatesofdiatomicmolecules, Phys.Rev.75(1949)29–31.

[15]F.Finkel,A. Gonzalez-Lopez,M.A.Rodriguez, Onthe familiesoforthogonal polynomialsassociatedtotheRazavypotential,J.Phys.A,Math.Gen.32(1999) 6821–6835.

[16]J.M. Mateos-Guilarte,J.M.Muñoz-Castañeda,A.Moreno-Mosquera,On super-symmetricDirac-deltainteractions,Eur.Phys.J.Plus130(2015)48.

[17]L.G. Mardoyan, G.S. Pososyan, A.N. Sisakyan, V.M. Ter-Antonyan, Two-dimensionalhydrogenatom.I.Ellipticbasis,JINR61(1984)1021–1024.

[18]C.P.Boyer,E.G.Kalnins,W.MillerJr.,Lietheoryandseparationofvariables.7. TheharmonicoscillatorinellipticcoordinatesandIncepolynomials,J.Math. Phys.16(1975)513–517.

[19]E.G. Kalnins, W.Miller Jr.,G.S. Pogosyan, Exactand quasi-exact solvability oftwo-dimensionalsuperintegrablequantumsystems,arXiv:math-ph/0412035, 2004.

[20]C. Cohen-Tannoudji,Bernard Diu,FranckLaloë,Quantum Mechanics, Wiley-Interscience,NewYork,1991.

[21]A. Galindo,P. Pascual,Quantum Mechanics,Springer,Berlin andNew York, 1990.

1. Explicit expressions for

E

and

J

at first order: Coulomb case

We have to specify the form of the perturbations

U

1

(q) and Φ

1

(q). For

the Coulomb approximation

µ <<

1, the natural split of

U

(q) in (42), main

text, into

U

0

(q) plus the perturbation

U

1

(q) is

U

0

(q) =

−

2

e

2

µ

1

cosh

α

−

cos

β

,

U

1

(q) =

e

2

_h

2

_µ

2

4

m

(sinh

2

_α

₋

_sin

2

_β

₎

_.

₍₁₎

Clearly if

µ <<

1,

U1

(q) is much smaller than

U0

(q). The split for Φ(q) in

(43), main text, is also rather obvious:

Φ

0

(q) =

−

me

2

_µ

2

cosh

α

cos

β

−

1

cosh

α

−

cos

β

,

Φ

1

(q) =

e

2

_h

2

_µ

4

16

sinh

2

α

sin

2

β .

(2)

In this case, the perturbation parameter is given by

=

µ

3

. If we redefine

e

e

2

_:=

_e

2

_µ

_{, where}

_e

_{is the charge, we have:}

H

1

=

e

e

2

h

2

4

m

(sinh

2

α

−

sin

2

β

)

,

T

1

=

e

e

2

h

2

16

sinh

2

α

sin

2

β .

(3)

In above equations we have used directly elliptic coordinates. Therefore,

all ingredients in the integrals resulting from (41), main text, have to be

written in terms of these coordinates. The differential

dx dy

should also be

changed according to the Jacobi theorem as follows:

dx dy

7−→

µ

2

4

(sinh

2

α

+ sin

2

β

)

dα dβ .

(4)

In (41), main text, the expression for Ψ

0

is the given for

ψ

n,r

in Tables 1

and 2. For higher values of

n

, we should use the procedure in the standard

bibliography. See References [11,12,15,16], main text.

Using the above equations and the wave function

ψ01

of table 1, it can be

shown that

h

ψ

01

|

ψ

01

i

=

µ

2

4

[

R

(1,0)

S

(0,0)

+

R

(0,0)

S

(1,0)

] =

πµ

2

2

ζ

2 0

,

(5)

where

ζ

0

=

m

e

e

2

_{. In turn}

h

ψ01

|

H1

|

ψ01

i

=

µ

2

4

e

e

2

_h

2

2

m

R

(2,0)

(

ζ0

)

S

(0,0)

(

ζ0

)

−

R

(0,0)

(

ζ0

)

S

(2,0)

(

ζ0

)

.

(6)

E

₁(0)

=

h

ψ

01

|

H

1

|

ψ

01

i

h

ψ

C

01

|

ψ

01C

i

=

e

e

2

_h

2

_ζ

2 0

4

πm

R

(2,0)

(

ζ0

)

S

(0,0)

(

ζ0

)

−

R

(0,0)

(

ζ0

)

S

(2,0)

(

ζ0

)

.

(7)

For the second constant of motion, we obtain

h

ψ01

|

T1

|

ψ01

i

=

e

e

2

_h

2

_µ

2

64

R(2

,0)

(

ζ0

)

S(1

,0)

(

ζ0

) +

R(1

,0)

(

ζ0

)

S(2

,0)

(

ζ0

)

.

(8)

Therefore,

J

(0) 1

=

e

e

2

_h

2

_ζ

2 0

32

π

R

(2,0)

(

ζ

0

)

S

(1,0)

(

ζ

0

) +

R

(1,0)

(

ζ

0

)

S

(2,0)

(

ζ

0

)

.

(9)

For

n

= 1, the solutions are unphysical as it was shown in last section,

and consequently we have to discard them. Thus, consider

n

= 2 and

h

ψ

21

|

ψ

21

i

=

−

µ

2

4

R

(2,0)

(

ζ

2

)

S

(1,0)

(

ζ

2

) +

R

(1,0)

(

ζ

2

)

S

(2,0)

(

ζ

2

)

,

(10)

expression which cannot be reduced. Then

h

ψ

21

|

H

1

|

ψ

21

i

=

−

µ

2

4

e

e

2

h

2

2

m

R

(3,0)

(

ζ

2

)

S

(1,0)

(

ζ

2

)

−

R

(1,0)

(

ζ

2

)

S

(3,0)

(

ζ

2

)

=

−

15

π

e

e

2

_h

2

_µ

2

mζ

6 2

.

(11)

Then,

E

₁(2,1)

=

h

ψ21

|

H1

|

ψ21

i

h

ψ

21

|

ψ

21

i

=

60

π

e

e

2

_h

2

mζ

6 2

1

R

(2,0)

(

ζ

2

)

S

(1,0)

(

ζ

2

) +

R

(1,0)

(

ζ

2

)

S

(2,0)

(

ζ

2

)

.

(12)

The second constant of motion takes the form

J

(2,1) 1

=

h

ψ

21

|

T

1

|

ψ

21

i

h

ψ21

|

ψ21

i

=

e

e

2

_h

2

16

R

(3,0)

(

ζ

2

)

S

(2,0)

(

ζ

2

) +

R

(2,0)

(

ζ

2

)

S

(3,0)

(

ζ

2

)

R

(2,0)

(

ζ2

)

S

(1,0)

(

ζ2

) +

R

(1,0)

(

ζ2

)

S

(2,0)

(

ζ2

)

.

(13)

For the wave functions

ψ

22

and

ψ

23

we have to use the relations in Part

3 at the end of the present Supplementary Material, so as to obtain:

E

₁(2,2)

=

h

ψ

22

|

H

1

|

ψ

22

i

h

ψ22

|

ψ22

i

=

e

e

2

_h

2

2

m

R(2

,2)

(

a

−

, ζ2

)

S(0

,2)

(

a

−

, ζ2

)

−

R(0

,2)

(

a

−

, ζ2

)

S(2

,2)

(

a

−

, ζ2

)

R

(1,2)

(

a

−

, ζ2

)

S

(0,2)

(

a

−

, ζ2

) +

R

(0,2)

(

a

−

, ζ2

)

S

(1,2)

(

a

−

, ζ2

)

(14)

and

J

(2,2) 1

=

h

ψ

22

|

T

1

|

ψ

22

i

h

ψ

22

|

ψ

22

i

=

e

e

2

_h

2

16

R

(2,2)

(

a

−

, ζ2

)

S

(1,2)

(

a

−

, ζ2

) +

R

(1,2)

(

a

−

, ζ2

)

S

(2,2)

(

a

−

, ζ2

)

R

(1,2)

(

a

−

, ζ

2

)

S

(0,2)

(

a

−

, ζ

2

) +

R

(0,2)

(

a

−

, ζ

2

)

S

(1,2)

(

a

−

, ζ

2

)

(15)

Finally, for

ψ

23

the result is identical to the last two equations, after the

replacement of

a

−

by

a

+

:

E

₁(2,3)

=

h

ψ

23

|

H

1

|

ψ

23

i

h

ψ23

|

ψ23

i

=

e

e

2

_h

2

2

m

R

(2,2)

(

a

+

, ζ

2

)

S

(0,2)

(

a

+

, ζ

2

)

−

R

(0,2)

(

a

+

, ζ

2

)

S

(2,2)

(

a

+

, ζ

2

)

R(1

,2)

(

a+, ζ2

)

S(0

,2)

(

a+, ζ2

) +

R(0

,2)

(

a+, ζ2

)

S(1

,2)

(

a+, ζ2

)

(16)

and

J

(2,3) 1

=

h

ψ

23

|

T

1

|

ψ

23

i

h

ψ23

|

ψ23

i

=

e

e

2

_h

2

16

R

(2,2)

(

a+, ζ2

)

S

(1,2)

(

a+, ζ2

) +

R

(1,2)

(

a+, ζ2

)

S

(2,2)

(

a+, ζ2

)

R

(1,2)

(

a+, ζ2

)

S

(0,2)

(

a+, ζ2

) +

R

(0,2)

(

a+, ζ2

)

S

(1,2)

(

a+, ζ2

)

(17)

From (12), (14) and (15), we observe that the degeneracy of the first

excited level is broken, since

E

₁(2,1)

6

=

E

₁(2,2)

6

=

E

₁(2,3)

which gives rise to three

single energy levels. The same becomes true with all other energy levels.

2. Explicit expressions for

E

and

J

at first order: the harmonic

oscillator

In the harmonic oscillator approximation,

µ >>

1, the roles of the

func-tions in (1) and (2) are obviously interchanged. Let us use ˘

e

=

eµ

2

. This

gives

U

0

(q) =

4

m

(sinh

2

α

−

sin

2

β

)

,

U

1

(q) =

−

µ

3

_cosh

_α

₋

_cos

_β

,

(18)

Φ

0

(q) =

˘

e

2

_h

2

16

sinh

2

α

sin

2

β ,

Φ

1

(q) =

−

m

e

˘

2

2

µ

3

cosh

α

cos

β

−

1

cosh

α

−

cos

β

.

(19)

Now, the perturbation parameter is

=

µ

−3

_{. Then,}

_H

1

and

T

1

in (36),

main text, have the following explicit form:

H

1

=

−

2˘

e

2

1

cosh

α

−

cos

β

,

T

1

=

−

m

e

˘

2

2

cosh

α

cos

β

−

1

cosh

α

−

cos

β

.

(20)

In order to perform the integrals which correspond to the first order of

approximation (41), main text, we write the measure (4) in the following

form:

dx dy

7−→

µ

2

4

(cosh

2

_α

₋

_cos

2

_β

₎

_{dα dβ}

₍₂₁₎

=

µ

2

4

(cosh

α

−

cos

β

)(cosh

α

+ cos

β

)

dα dβ .

Then, some singular denominators cancel out and the integrals should be

quite similar as in the first case. With the help of the integrals

V

j

and

W

j

, we

obtain the following results (here the functions

ψ

n,r

are those corresponding

to the harmonic oscillator approximation):

h

ψ

01

|

ψ

01

i

=

µ

2

4

[

V

2

W

0

+

V

0

W

2

]

,

(22)

so that

h

ψ01

|

H1

|

ψ01

i

=

−

µe

2

2

[

V1W0

+

V0W1

]

(23)

and

h

ψ

01

|

Φ

1

|

ψ

01

i

=

−

me

2

µ

3

8

[(

V

2

−

V

0

)

W

1

+ (

W

2

−

W

0

)

V

1

]

(24)