Physics Letters A 306 (2003) 291–295

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Quasi exactly solvable operators and Lie superalgebras

Yves Brihayea, Betti Hartmannb,∗

a Faculté des Sciences, Université de Mons-Hainaut, B-7000 Mons, Belgiumb Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK

Received 5 November 2002; received in revised form 18 November 2002; accepted 20 November 2002

Communicated by J.P. Vigier

Abstract

Linear operators preserving the direct sum of polynomial ringsP(m) ⊕ P(n) are constructed. In the case|m − n| = 1they correspond to atypical representations of the superalgebra osp(2,2). For |m − n| = 2 the generic, finite-dimensionalrepresentations of the superalgebraq(2) are recovered. An example of a Hamiltonian possessing such a hidden algebra isanalyzed. 2002 Elsevier Science B.V. All rights reserved.

PACS: 02.20.Sv; 03.65.Fd

1. Introduction

Quasi exactly solvable (QES) operators [1] refer tolinear differential operators which preserve a finite-dimensional vector space of smooth functions. Theycan be used in the framework of quantum mechan-ics to construct Hamiltonians which possess a finitenumber of algebraic eigenfunctions. The classifica-tion of QES operators constitutes an interesting mathe-matical problem which generalizes the Bochner prob-lem [2].

Having chosen the vector spaceV of functions to beleft invariant, the first trial is to construct a set of basicoperators preservingV from which all others can begenerated in the sense of an enveloping algebra. The

* Corresponding author.E-mail addresses: yves.brihaye@umh.ac.be (Y. Brihaye),

betti.hartmann@durham.ac.uk (B. Hartmann).

fact of having a set of normal ordering rules for thebasic operators is also crucial at this stage.

However, one of the striking aspects of QES oper-ators is their close relation to the theory of representa-tions of Lie algebras and superalgebras. In particular,the realisations of Lie algebras in terms of differentialoperators play a crucial role [3].

Of course, the fact that QES operators are particularrealisations of Lie algebras is not a necessary require-ment, as shown in [4], but it constitutes an advantagebecause, if so, representation theory helps construct-ing families of invariant vector spaces. In this Letter,we put emphasis on the realisations of the algebraq(2)in terms of QES operators. We discuss some gener-alities in Section 2 and give our construction in Sec-tion 3. An example of a QES Schrödinger equation re-lated to these operators is analyzed in Section 4. Weend with concluding remarks and an outlook in Sec-tion 5.

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01636-5

292 Y. Brihaye, B. Hartmann / Physics Letters A 306 (2003) 291–295

2. Generalities

Let us denote byP(n) the ring of polynomials ofdegree less or equal ton in a real variablex. The mainresult concerning the construction of QES operators[1] is the discovery that the linear operators whichpreserveP(n) constitute the enveloping algebra of theLie algebrasl(2) in the representation

j+n = x(D − n), j0

n = D − n

2,

(1)j− = d

dxwith D ≡ x

d

dx.

Extending this result to the case of linear operatorswhich preserve the vector spaceP(n − ∆) ⊕ P(n) itwas shown in [4] that these operators can be assembledfrom 2∆ + 6 basic operators which can be chosenaccording to

T + =(j+(n−∆) 0

0 j+n

), T 0 =

(j0(n−∆) 0

0 j0n

),

(2)T − =(j− 00 j−

),

(3)J = 1

2

(n+∆ 0

0 n

).

The three operatorsT ±,0 obey thesl(2) algebra:

(4)[T +, T −]= −2T 0,

[T ±, T 0]= ∓T ±

andJ commutes with allT ’s. In the following theseoperators play the role of bosonic operators. They haveto be completed by 2(∆+ 1) off-diagonal ones:

(5)Qα = xα−1σ−, α = 1, . . . ,∆+ 1,

(6)Qα = qα,(n)σ+, α = 1, . . . ,∆+ 1

with the definition

qα,(n) =(∆−α∏j=0

(D − (n+ 1−∆)− j

))

(7)×(

d

dx

)α−1

andσ± = (σ1 ± iσ2)/2.

2.1. Commutation relations

After an algebra, the commutation relations be-tween the diagonal operators and the off-diagonal onescan be obtained [4] :

(8)[T +,Qα

]= −(1− α +∆)Qα+1,

(9)[T 0,Qα

]= −(

1− α + ∆

2

)Qα,

(10)[T −,Qα

]= −(1− α)Qα−1,

(11)[T +,Qα

]= (1− α)Q1−α,

(12)[T 0,Qα

]=(

1− α + ∆

2

)Qα,

(13)[T −,Qα

]= (1− α +∆)Qα+1.

Using the representations ofsl(2), these formulaereveal that the set of operatorsQ (and independentlyQ) transforms according to the representation of spins ≡ ∆/2 under the adjoint action ofT ’s (they are alsocalled tensorial operators). In fact bothQα andPα ≡Q∆+2−α behave exactly the same under theT ’s. Interms of Young diagrams, the representation generatedby theQ’s (or theP ’s) is characterized by the Youngdiagram with one line of∆ boxes.

On the other hand,J behaves as a grading operator:

(14)[J,Qα] = −∆

2Qα,

[J,Qα

]= ∆

2Qα.

One can then convince oneself that the most naturalset of ordering rules is obtained whenanticommuta-tors between theQ’s and theQ’s are chosen, whichmakes the interpretation of theQ’s andQ’s as fermi-onic operators natural. The unpleasant feature aboutthe algebraic structure generated byT ,J,Q,Q is thatthe anticommutators{Qα,Qβ} are generically poly-nomials of degree∆ in the bosonic operators. In thecase∆ = 1 the operators constitute an atypical repre-sentation of the superalgebra osp(2,2) [3]. For ∆> 1the operatorsT ,Q,Q do not seem to be related to Liesuperalgebras. However, we will show in the followingsection that it is possible to find a relation for∆ = 2.

3. The case ∆ = 2

We now consider the case∆ = 2. This case has thepeculiarity that, under theT ’s, the operatorsQ and the

Y. Brihaye, B. Hartmann / Physics Letters A 306 (2003) 291–295 293

P transform according to the adjoint representation, inother words they transform as triplets ofsl(2). Let uscharacterize a tripletV1,V2,V3 by[T +,Vα

]= (1− α)Vα−1,[T 0,Vα

]= (2− α)Vα,

(15)[T −,Vα

]= (3− α)Vα+1

with α = 1,2,3. The following sets of operators

(16)Qα ≡ (Q1,Q2,Q3

),

(17)Pα ≡ (Q3,Q2,Q1),

(18)Tα ≡ (T +, T 0, T −)≡ (T1, T2, T3),

therefore, transform as triplets under theT ’s. Obvi-ously any linear combination of these triplets (withcoefficients being operators commuting with theT ’s)also constitutes a triplet.

The case∆ = 2 in this sense in special sincefor ∆ > 2 no match between the fermionic andthe bosonic operators seems to exist. This is alsotrue for QES operators depending on many variables[5]. In this case the diagonal operators obey thecommutation relations ofsl(V + 1) with V being thenumber of independent variables. The counterpart oftheQ’s then corresponds to the completely symmetricrepresentation ofsl(V + 1) with a Young diagramcontaining one line with∆ boxes. In contrast, theadjoint representation ofsl(V + 1) is characterized bya diagram withV − 1 lines, the first line with twoboxes, the others with one box. The caseV = 1,∆ = 2is, therefore, very peculiar.

To proceed further we remind that in the casestudied here the anticommutators ofQ with Q leadto quadratic polynomials in theT ’s andJ , the detailsof which are given in [4]. In order to obtain amore conventional algebra, we take advantage of thecoexistence of three independent triplets of operatorsand try to simplify the algebra of operators preservingP(n − 2) ⊕ P(n). For this purpose, we define a newtriplet of operators,Fα :

(19)Fα ≡ Qα + cPα +DTα, α = 1,2,3,

wherec is a constant andD is a constant diagonalmatrix. If we chooseD2 = 12 andc = −1 we obtain

(20){Fα,Fβ } = n2gαβ,

where gαβ is the Cartan metric ofsl(2). In therepresentation used, the non-zero elements of thismetric are given by

(21)gαβ = 1 if α,β = 1,3 or 3,1

(22)= −1

2if α = β = 2.

From (20) it is apparent that the anticommutators{Fα,Fβ} are now linear combinations of the bosonicoperators.

To establish this result, the identities

(23)jβ,(n)pα − pαjβ,(n−2) = (β − α)pα+β−2,

(24)

jβ,(n−2)qα,(n) − qα,(n)jβ,(n) = (β − α)qα+β−2,(n),

have to be used, and we introduced the notationspα ≡x3−α andj1,(n) ≡ j+

n , j2,(n) ≡ j0n , j3,(n) ≡ j−.

As an alternative to the operators (2)–(6) above, wepropose the set given byT1, T2, T3 and completed by

Fα ≡ 1√nFα, α = 1,2,3,

(25)h0 = n12, h1 = √nσ3.

Now, T ±,0, h0 are the bosonic operators andFα,h1the fermionic operators. They fulfill the (anti)commu-tation relations of the superalgebraq(2) [6]. In partic-ular, we find

(26){Fα, Fβ

}= gαβh0,

(27){Fα, h1

}= 2Tα,

(28){h1, h1} = 2h0.

This latter set of operators therefore constitutes aseries of realisations of the superalgebraq(2) byQES operators. This series is labelled by an inte-gern and the 2n-dimensional vector space preservedis P(n− 2)⊕P(n). Notice that this result was pre-sented in [6], but the calculation was done by “bruteforce”. Our derivation uses the representation structureof the operatorsQ andQ and, moreover, demonstratesthe importance of the case∆ = 2.

4. Example of a QES Hamiltonian for ∆ = 2

In this section, we discuss examples of QES Schrö-dinger operators preserving the vector spaceP(n) ⊕

294 Y. Brihaye, B. Hartmann / Physics Letters A 306 (2003) 291–295

P(n − 2) and construct the eigenvalues forn = 2 andn = 3. The Hamiltonian is given by:

H = − d2

dy212 + y612 + (1− 4n)y212

(29)− 4y2σ3 − 4nk0σ1,

where σ1, σ3 are the Pauli matrices,k0 is an arbi-trary constant andn an integer. To our knowledge, thisHamiltonian is the only possible QES matrix Hamil-tonian with polynomial potential [7,8] (which can,however, be generalised through the inclusion of ay4

term). We introduce the following change of basis andvariable:

H = U−1HU with

(30)U = e−x2/4(

1 0k0

ddx

1

), x = y2.

Then, the Hamiltonian reads:

H = −(

4xd2

dx2 + 2d

dx

)12 − 4nk2

0d

dxσ3

+ 4

(x2 d

dx− nx 0

0 x2 ddx

− (n− 2)x

)(31)+ 4k0

(0 −n

(1+ k20n)

d2

dx2 0

)which obviously preservesP(n)⊕P(n− 2).

As a consequence, 2n algebraic eigenvectors ofH can be constructed. In the limitk0 = 0 twodecoupled sextic QES Hamiltonians are recovered. Inthe following, we discuss the algebraic spectrum ofH

for n = 2 andn = 3.Forn = 2 the four eigenvalues are given by

(32)E2 = 32+ c2 ± 4√

64+ 2c2, c ≡ −4nk0

and the corresponding eigenvectors are

(33)φ ∝(y4 − E

4 y2 + E2−c2−48

32

− c4y

2 + E(E2−c2−64)32c

)e−y4/4.

The degeneracy of the energy levels forc = 0 withE = −8,0,0,8 is lifted for c > 0. In the limit c → 0,two of the eigenvectors (33) converge to linear com-binations of the two eigenvectors of the decoupledsystem withE = 0. The ground state energy corre-

sponds toE(0) = −√

32+ c2 + 4√

64+ 2c2 and both

Fig. 1. The modulus of the energyeigenvalues is shown for the QESHamiltonian presented in Section 4 withn = 3.

components of the corresponding eigenvectorφ haveno nodes. Denoting the four algebraic energy levelsby E(a), a = 0,1,2,3, in increasing order and byk1(a), k2(a) the number of nodes of the two compo-nents of the corresponding eigenvectorφ(a), we ob-tain:

a = 0 a = 1 a = 2 a = 3

k1(a) 0 2 2 4k2(a) 0 2 0 2

For n = 3, the six algebraic eigenvalues are thesolutions of the equation

E6 − (248+ 3c2)E4 + (

4800+ 240c2 + 3c4)E2

(34)− (23040− 1344c2 − 8c4 + c6)= 0.

As for the casen = 2 it is apparent that the spectrum ofthe equation is invariant under the reflectionE → −E.The three values of|E| are plotted as functions ofcin Fig. 1. This demonstrates in particular that a leveldegeneracy occurs atc ≈ 5.

The reflection symmetry of the energy eigenvaluesE → −E can be demonstrated for arbitrary values ofn by using a similar argument as pointed out in [9]for scalar equations. In the case of (29) the relevantsymmetry of the eigenvalue equation is:

(35)y → iy, φ(E) → σ3φ(−E).

Y. Brihaye, B. Hartmann / Physics Letters A 306 (2003) 291–295 295

Of course, as in the case of [9], the inclusion of they4-term in the potential would spoil this reflectionsymmetry.

To our knowledge, no QES matrix Schrödingeroperator is known which preservesP(n) ⊕P(n − ∆)

with ∆ � 3.

5. Concluding remarks and outlook

In this Letter we have given a new constructionof the realisations of the super Lie algebraq(2) bymeans of QES operators. This demonstrates that theset of operators preserving the vector spaceP(n−2)⊕P(n) is just isomorphic to the enveloping algebra ofq(2). The quadratic algebra calledA(2) in [3] can bereplaced byq(2).

The natural extension of these results would be tostudy∆ > 2. Unfortunately, for generic values of∆the combination (19) is limited toFα = Qα + cQα ,(α = 1, . . . ,∆ + 1) and no simplification occurs. Foreven values of∆, multiplets having the same tensorialstructure as theQ’s can be constructed out of theT ’s,e.g., for∆ = 4 we find:

Sα =((

T +)2, 12

{T +, T 0}, 1

3

(2(T 0)2 + 1

2

{T +, T −}),

(36)12

{T 0, T −}, (T −)2)

which behaves as a 5-plet (similarly to theQα ’s undertheT ’s). Considering combinations analog to (19)

(37)Fα = Qα + cPα +DSα, α = 1, . . . ,5

we have shown that no values ofc and D can beconstructed such that the anticommutators{Fα,Fβ}are polynomials in the bosonic operators.

Another possible extension of these considerationswould be the construction of finite difference operatorspreservingP(n − 2) ⊕ P(n). It could be checkedthen, whether such operators obey some deformationof q(2) in a similar way as in [3] where the relevantstructure is the so-called quommutator deformation ofthe Lie superalgebra osp(2,2).

Acknowledgements

Y.B. gratefully acknowledges the Belgian FNRSfor financial support. B.H. was supported by an EP-SRC grant.

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