otential isspectrumation

Physics Letters A 340 (2005) 31–36

www.elsevier.com/locate/pla

Double shape invariance of the two-dimensional singularMorse model

F. Cannataa, M.V. Ioffe b,∗, D.N. Nishnianidzeb,c

a Dipartimento di Fisica and INFN, Via Irnerio 46, 40126 Bologna, Italyb Sankt-Petersburg State University, 198504 Sankt-Petersburg, Russia

c Kutaisi Technical University, 4614 Kutaisi, Republic of Georgia

Received 23 December 2004; received in revised form 16 March 2005; accepted 23 March 2005

Available online 8 April 2005

Communicated by P.R. Holland

Abstract

A second shape invariance property of the two-dimensional generalized Morse potential is discovered. Though the pnot amenable to conventional separation of variables, the above property allows to build purely algebraically part of theand corresponding wave functions, starting fromonedefinite state, which can be obtained by the method of SUSY-separof variables, proposed recently. 2005 Elsevier B.V. All rights reserved.

PACS:03.65.-w; 03.65.Fd; 11.30.Pb

Keywords:Supersymmetrical quantum mechanics; Shape invariance; Partial solvability

hari-hat,

en-allyngel.ralo-nce

onee-acy

1. Introduction

Recently the notion of shape invariance[1] wasgeneralized[2,3] to two-dimensional cases, whicare not amenable to conventional separation of vables in the Schrödinger equation. It was shown t

* Corresponding author.E-mail addresses:cannata@bo.infn.it(F. Cannata),

m.ioffe@pobox.spbu.ru(M.V. Ioffe), qutaisi@hotmail.com(D.N. Nishnianidze).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.03.071

in contrast to the one-dimensional situation, in geral the shape invariance itself gives algebraiconly part of spectra (and wave functions), leadito the partial (quasi-exact) solvability of the modThe fact that only a partial solution for the spectproblem is provided by shape invariance for twdimensional problems is related to the depende(in general) of theground state energyon the pa-rameters of the model (in one-dimensional caseusually hadEgs ≡ 0, not depending on the paramter of shape invariance). Also, possible degener

.

32 F. Cannata et al. / Physics Letters A 340 (2005) 31–36

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in-

wsof

es itearhento

nce

ofarepeand to

layin-

tond

he

ls.po-ne

-

hy-

gu-

n-

i-Its:

u-

slyti-

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adees.

of levels in two-dimensional models is important: fcomplete solvability one has to know both energy lels andall corresponding wave functions.

In particular, the two-dimensional generalized (sgular1) Morse potential was demonstrated[2,3] to sat-isfy shape invariance. Furthermore, this model allofor SUSY-separation of variables (i.e., separationvariables in the supercharge), and therefore makpossible to construct some wave functions as lincombinations of zero modes of the supercharge. Teach of these “principal” eigenfunctions was usedgenerate (purely algebraically!) the shape invariachain of excited levels.

In the present Letter some additional propertiesthe two-dimensional generalized Morse potentialstudied (Section2). It is shown that besides the shainvariance, described in[2], the model possessesadditional shape invariance, which can also be useconstruct corresponding chains of states (Section3).In this construction the wave functions of thefirstshape invariance chain of the previous section pthe role of the “principal” states for new shapevariance chains. Thus it is shown (Section4) thatthe combination of both shape invariances allowsbuild algebraicallythe same part of the spectrum acorresponding eigenfunctions starting fromonly one“principal” state. The mutual interrelation between t“old” and “new” chains is also clarified.

2. Shape invariant two-dimensional Morse model

The directd-dimensional(d � 2) generalizationof Witten’s SUSY QM includes[4] Schrödinger op-erators both with scalar and with matrix potentiaIn order to avoid the appearance of matrix comnents in the two-dimensional superhamiltonian, ocan explore[3,5] for the cased = 2 the idea of onedimensional higher order SUSY QM[6] and to take,for example, the second order supercharges withperbolic (Lorentz) metrics:

Q± = ∂21 − ∂2

2 + Ci(�x)∂i + B(�x),

(1)�x = (x1, x2), ∂i = ∂/∂xi, i = 1,2.

1 We remark that the behaviour of wave functions at the sinlarity is under control (see[2]).

The main relations of SUSY QM are the two-dimesional SUSY intertwining relations:

H(1)Q+ = Q+H(2),

(2)H(1,2) = −�(2) + V (1,2)(�x), �(2) ≡ ∂2i ,

for which some partial solutions were found[5] pro-viding nontrivial models with Schrödinger HamiltonansH(1), H (2), not allowing separation of variables.is convenient to consider also light-cone coordinate

x± ≡ x1 ± x2, ∂± = 1

2(∂1 ± ∂2),

(3)C± = C1 ∓ C2,

where[5] due to(2) C± = C±(x±), and

∂−(C−F) = −∂+(C+F),

(4)F(�x) ≡ F1(2x1) + F2(2x2).

The potentialsV (1,2)(�x) are:

V (1,2)(�x) = ±1

2

(C′+(x+) + C′−(x−)

)

+ 1

8

(C2+(x+) + C2−(x−)

)

(5)+ 1

4

(F2(2x2) − F1(2x1)

),

and

(6)B(�x) = 1

4

(C+C− + F1(2x1) + F2(2x2)

).

Recently two specific models of the list of particlar solutions of(4) were shown[2,7] to be partially(quasi-exactly[8]) solvable, i.e., part of eigenvalueand eigenfunctions of the system was found anacally.

The first of the models, the two-dimensional Morpotential, is characterized2 by

C+(x+) = 4aα,

(7)C−(x−) = 4aα coth

(αx−

2

),

1

4F1,2(2x1,2) = ∓(2c − α)exp(−αx1,2)

(8)∓ exp(−2αx1,2).

2 The constants used here differ from those in[2,3]. The presentchoice is useful for the derivation below and can be easily mconsistent with the previous ones by suitable shifts of coordinat

F. Cannata et al. / Physics Letters A 340 (2005) 31–36 33

sin-rse

ialra-re-

ot insy-m-

-s of

-

in-

ce,

ofe-

i-

ec-n-,

r-ted

With this choice the potentials(5) can be naturallyinterpreted as a two-dimensional (non-separable,gular) generalization of the one-dimensional Mopotential:

V (1,2)(�x;a,α, c)

= α2a(2a ∓ 1)sinh−2(αx−/2)

+ (2c − α)(exp(−αx1) + exp(−αx2)

)(9)+ (

exp(−2αx1) + exp(−2αx2)) + 4a2α2.

While the Schrödinger equations with potentV (1,2) in (9) are not amenable to standard sepation of variables, nevertheless, we can apply thecently proposed[2] (see also[3]) method of SUSY-separation of variables (variables are separable nH(1,2), but in the superchargeQ+). Then zero modeof Q+ were constructed analytically in terms of hpergeometric functions: suitably chosen linear cobinations of them provide (see details in[2,3]) a setof “principal” wave functionsΨ (2)

k,0(�x;a,α, c) of the

Hamiltonian H(2)(�x;a,α, c) and corresponding energy eigenvalues, which depend on arbitrary valueparameters:

E(2)k,0(a,α, c) = −2

[2aα2sk − εk

]

= −2

[−2aα2

(c

α+ k

)+ (c + kα)2

],

(10)k = 0,1,2, . . . .

εk andsk above were defined in[2], and are now expressed in terms of new parameterc instead ofA ≡(c − α/2)2 (the appropriate constant shift ofx1,2 wasalso used here).

The model of Eq.(9) enjoys [2] an additionalremarkable property—the two-dimensional shapevariance:

H(1)(�x;a,α, c) = H(2)(�x;a − 1

2, α, c) +R(a,α, c),

(11)R(a,α, c) = α2(4a − 1).

Similarly to the one-dimensional shape invarianeach “principal” eigenfunctionΨ (2)

k,0 gives start toa whole shape invariance chain of eigenstatesH(2)(�x;a,α, c), which can be built by means of a squence of superchargesQ− ≡ (Q+)† :Ψ

(2)k,m(�x;a,α, c)

= Q−(a,α, c)Q−(a − 1, α, c

)

2

× Q−(a − 1, α, c) · · ·Q−(a − 1

2(m − 1), α, c)

(12)× Ψ(2)k,0

(�x;a − 12m,α, c

)(m = 1,2, . . .),

with eigenvalues:

E(2)k,m(a,α, c)

= E(2)k,0

(a − 1

2m,α, c)

+R(a − 1

2(m − 1), α, c) + · · · +R(a,α, c)

= 2(c + kα)(2aα − aαm − (c + kα)

)(13)+ α2m(4a − m).

The sequence in(12) is constrained only by the condtion of normalizability of these wave functions.

3. The new shape invariance of the Morsepotential

In order to demonstrate the existence of a sond invariance of the two-dimensional Morse potetial (9), we link it to another two-dimensional systemfound in [5] among solutions(4)–(6) of intertwiningrelations with hyperbolic (Lorentz) metrics in supecharges. The corresponding functions will be denoby tildes and their arguments by�y = (y1, y2):

C+(y+) = 4(exp(αy+) + c

),

C−(y−) = 4(exp(αy−) + c

), y± ≡ y1 ± y2,

F1(2y1) = 0,

1

4F2(2y2) = 2d

(exp(αy2) − exp(−αy2)

)−2,

B = 4(exp(αy−) + c

)(exp(αy+) + c

)+ 2d sinh−2(αy2).

The superpartner potentials found in[5]:

V (1,2)(�y;a,α, c)

= 2(exp(2αy+) + exp(2αy−)

)+ 2(2c ± α)

(exp(αy+) + exp(αy−)

)(14)+ 2d sinh−2(αy2),

satisfy to the intertwining relations:

H (1)(�y;a,α, c)Q+(�y;a,α, c)

= Q+(�y;a,α, c)H (2)(�y;a,α, c),

34 F. Cannata et al. / Physics Letters A 340 (2005) 31–36

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tri-, bu

.

in-ye

it

if-s

ers

ari-re-owsing.

ex-ein

note-rdly

this

edin-theof

ri-tinge

Q+(�y;a,α, c)

= ∂2y1

− ∂2y2

+ C+(y+)∂y− + C−(y−)∂y+ + B.

In order to relate this system to the model of Stion 2, one has to replace:

y+ ≡ −x1, y− ≡ −x2, d ≡ α2a(2a + 1),

H (1,2)(�y(�x);a,α, c

) ≡ 2h(1,2)(�x;a,α, c),

Q+(�y(�x);a,α, c) ≡ q+(�x;a,α, c).

The componentsh(1,2)(�x;a,α, c) of new super-Ha-miltonian become:

h(1,2)(�x;a,α, c)

= −�(2) + α2a(2a + 1)sinh−2(αx−/2) + 4a2α2

+ (2c ± α)(exp(−αx1) + exp(−αx2)

)(15)+ (

exp(−2αx1) + exp(−2αx2)),

and they are intertwined:

h(1)(�x;a,α, c)q+(�x;a,α, c)

(16)= q+(�x;a,α, c)h(2)(�x;a,α, c)

by the supercharge:

q+(�x;a,α, c)

= 4∂1∂2 − 4(exp(−αx1) + c

)∂2

− 4(exp(−αx2) + c

)∂1

+ 4(exp(−αx1) + c

)(exp(−αx2) + c

)(17)+ 2α2a(2a + 1)sinh−2(αx−/2).

From(15) one can conclude that this supersymmecal system has also the shape invariance propertyin this case in the parameterc:

(18)v(1)(�x;a,α, c) = v(2)(�x;a,α, c + α),

where the term, analogous toR in (11), now vanishesTherefore also the spectrum ofh(2)(�x;a,α, c) canbe obtained algebraically: starting from some “prcipal” (for this model) wave function with energe(2)l,0(a,α, c), l = 0,1,2, . . . , one will obtain the shap

invariance chain of states with energies

e(2)l,n(a,α, c) = e

(2)l,0(a,α, c + nα),

(19)n = 1,2, . . . .

As for the choice of “principal” states for this model,is convenient to choose states of thefirst shape invari-ance chain of Section2: e

(2)(a,α, c) = E

(2)(a,α, c).

k,0 0,k

t

Comparing the two supersystems(9) and(15), onenotices that:

(20)H(2)(�x;a,α, c) = h(2)(�x;a,α, c),

therefore the same two-dimensional system(9)—withgeneralized Morse potential—participates in two dferent intertwining relations(2) and(16)and possessealso two independent shape invariance properties(11)and(18), expressed in transformations of parameta andc, respectively.

4. The spectrum of the singular Morse potential

A priori, one has to expect, that each shape invance will give rise to its own chain of states and corsponding energies. But a more careful analysis shthat these chains include states which are overlappFirst of all, one can observe this overlap from theplicit formulas(13) and(19) for the spectra. But duto the possible (in principle) degeneracy of levelstwo-dimensional quantum mechanics, it still doesimply the coincidence of wave functions. Neverthless, this coincidence holds as can be straightforwachecked by means of the operator equality:

q−(a,α, c)Q−(a,α, c + α)

= Q−(a,α, c)q−(a − 1

2, α, c),

(21)q−(�x;a,α, c) ≡ (q+(�x;a,α, c)

)†.

For example, just the operators in both sides ofequation, acting onto the wave functionΨ (2)

0,0 (a −12, α, c + α), give by two different waysthe eigen-

functionΨ(2)1,1 (a,α, c) with energyE(2)

1,1(a,α, c). Anal-

ogously, one can check that the wave functionΨ(2)k,m

with arbitrary pair of indices(k,m) can be obtainedby different ways, via chains of operatorsQ− andq−,giving the same result due to equalities similar to(21).

A better understanding of the above mentionoverlap can be obtained by realizing that the “prcipal” states of the new shape invariance (i.e.,states from which one starts the constructionchains) are chosen to be the statesE

(2)0,m(a,α, c) =

e(2)m,0(a,α, c) (see(13)). So, the second shape inva

ance acts transversely in respect to the first. Acwith q−(�x;a,α, c) (see(17)) k times on a generic stat

F. Cannata et al. / Physics Letters A 340 (2005) 31–36 35

in

ral

a-edhela-

o-

in-

ofediale

the

o-and

ainsed

ar-an

his

e-a-

ian

yl-

an05-

2)

)

.

.

.

Ψ(2)0,m(a,α, c) leads to the stateΨ (2)

k,m. Thus the overlapcan be concisely depicted as

(22)E(2)k,m = e

(2)m,k.

In other words, one can move first “up”c → c+α andthen “transversely”a → a − 1

2, or first transverselyand then up on the “energy lattice”.

5. Remark and conclusions

We stress that each Hamiltonian, participatingSUSY intertwining relations of second order(2), is in-tegrable[3,5], i.e., it has a symmetry operator (integof motion) of fourth order in derivatives:

R(1) = Q+Q−, R(2) = Q−Q+,

(23)[H(i),R(i)

] = 0, i = 1,2,

which cannot be reduced by elimination of opertor functions of the Hamiltonian. It can be checkstraightforwardly, though rather tediously, that tsymmetry operators associated to intertwining retions (16) do notgive a new,independent, symmetryoperator for the HamiltonianH(2), indeed:

q−q+ + Q−Q+

(24)= 4(H(2) + 2c2)2 + 16c2(c2 − 4a2α2).

The gluing condition(20)of two SUSY systems(9)and(15) provides an opportunity to link the compnentsH(1) andh(1) by intertwining operators of fourthorder in derivatives:

(25)H(1)(Q+q−) = (

Q+q−)h(1).

This intertwining produces also a shape invariancevolving two parameters:

H(1)(�x;a,α, c) = h(1)(�x;a − 1

2, α, c − α)

(26)+R(a,α, c).

The associated symmetry operators forH(1) of ordereight (Q+q−)(q+Q−) are reducibleto a polynomialfunction ofH(1) andR(1).

Let us note that treatments, similar to the onesSections2–4, can be applied also to the complexifiversion of singular two-dimensional Morse potent(9) (see[9]), including a complexified version of th

related two-dimensional model(14) and leading to acomplex form of double shape invariance.

In conclusion, the main results of the Letter arefollowing.

• A new property of two-dimensional Morse mdel—the second shape invariance—was foundinvestigated.

• The excited states of new shape invariance chwere proved to coincide with the states obtainfrom the first shape invariance.

• It was shown that the same set of states of ptially solvable two-dimensional Morse model cbe built now from only one “principal” stateΨ

(2)0,0 (a,α, c), i.e., one needsonly the firstzero

mode of the superchargeQ+ to be obtained bythe method of SUSY-separation of variables. Tis much easier to build.

• Though the Morse Hamiltonian(11) obeys twodifferent intertwining relations and has, corrspondingly, two fourth order symmetry opertors, the systemis not superintegrable, sincethese operators are interrelated by the hamilton(see(24)).

Acknowledgements

M.V.I. and D.N.N. are grateful to the Universitof Bologna and INFN for support and kind hospitaity. This work was partially supported by the RussiFoundation for Fundamental Research (Grant No.01-01090).

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