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Exactly Solvable Potentials and Quantum Algebras1

arXiv:hep-th/9112075v2 24 Jul 1992UdeM-LPN-TH75Revised versionExactly Solvable Potentials and Quantum Algebras1Vyacheslav Spiridonov2,3Laboratoire de Physique Nucl´eaire, Universit´e de Montr´eal,C.P. 6128, succ.

A, Montr´eal, Qu´ebec, H3C 3J7, CanadaAbstractA set of exactly solvable one-dimensional quantum mechanical potentials is de-scribed. It is defined by a finite-difference-differential equation generating in the lim-iting cases the Rosen-Morse, harmonic, and P¨oschl-Teller potentials.

General solutionincludes Shabat’s infinite number soliton system and leads to raising and lowering op-erators satisfying q-deformed harmonic oscillator algebra. In the latter case energyspectrum is purely exponential and physical states form a reducible representation ofthe quantum conformal algebra suq(1, 1).PACS numbers: 03.65.Fd, 03.65.Ge, 11.30.Na1Published in Phys.Rev.Lett., vol.69, n.3 (1992) 398-401.2On leave from the Institute for Nuclear Research, Moscow, Russia3E-mail address: spiridonov@lps.umontreal.ca

Lie algebras are among the cornerstones of modern physics. They have enormous numberof applications in quantum mechanics and, in particular, put an order in classification ofexactly solvable potentials.

”Quantized”, or q-deformed, Lie algebras (also loosely calledquantum groups) are now well established objects in mathematics [1]. Their applicationswere found in two-dimensional integrable models and systems on lattices.

However, despiteof much effort quantum algebras do not yet penetrate into physics on a large scale.Inthis paper we add to this field and show that a q-deformed harmonic oscillator algebra [2]may have straightforward meaning as the spectrum generating algebra of the specific one-dimensional potential with exponential spectrum. This result shows that group-theoreticalcontent of exactly solvable models is not bounded by the standard Lie theory.Recently Shabat analyzed an infinite chain of reflectionless potentials and constructed aninfinite number soliton system [3].

The limiting potential decreased slowly at space infinitiesand obeyed peculiar self-similar behavior. We will present corresponding results in slightlydifferent notations.

We denote space variable by x and introduce N superpotentials Wn(x)satisfying the following set of second order differential equations(W ′n + W ′n+1 + W 2n −W 2n+1)′ = 0,n = 0, . .

. , N −1(1)where primes denote derivative w.r.t.

x. Taking first integralsW ′n + W ′n+1 + W 2n −W 2n+1 = kn+1,(2)where kn are some constants, we define N + 1 Hamiltonians2Hn = p2 + Un(x),p ≡−id/dx,(3)U0(x) = W 20 −W ′0 + k0,Un+1(x) = Un(x) + 2W ′n(x).An arbitrary energy shift parameter k0 enters all potentials Un(x).Notorious supersymmetric Hamiltonians are obtained by unification of any two successivepairs Hn, Hn+1 in a diagonal 2 × 2 matrix [4].

Analogous construction for the whole chain(3) was called an order N parasupersymmetric quantum mechanics [5, 6].In the lattercase relations (1) naturally arise as the diagonality conditions of the general (N + 1) ×(N + 1)-dimensional parasupersymmetric Hamiltonian. We do not use here these algebraicconstructions and consider operators Hn on their own ground.If Wn(x)’s do not have severe singularities then the spectra of operators (3) may differonly by a finite number of lowest levels.

Under the additional condition that the functionsψ(n)0 (x) = e−R x Wn(y)dy(4)belong to the Hilbert space L2 one obtains first N eigenvalues of the Hamiltonian H0H0 ψ(0)n (x) = En ψ(0)n (x),En = 12nXi=0ki,n = 0, 1, . .

. , N −1,(5)1

where subscript n numerates levels from below. In this case (4) represents ground statewave function of Hn from which one can determine lowest excited states of Hj, j < n, e.g.,eigenfunctions of H0 are given byψ(0)n (x) ∝(p + iW0)(p + iW1) .

. .

(p + iWn−1) ψ(n)0 . (6)Any exactly solvable discrete spectrum problem can be represented in the form (2)-(6).Sometimes it is easier to solve Schr¨odinger equation by direct construction of the chainof associated Hamiltonians (3) – this is the essence of so called factorization method [7-9].

For the problems with only N bound states there does not exist WN(x) making ψ(N)0normalizable. For example, if WN(x) = 0, then Hn has exactly N −n levels, the potentialUn(x) is reflectionless and corresponds to (N −n)-soliton solution of the KdV-equation.Let us consider potentials which support infinite number of bound states, N = ∞.

In thiscase one can derive from (2) the following differential equations involving only one derivativeand a tail of Wn’sW ′i(x) + W 2i (x) +∞Xj=1(−1)j(2W 2i+j(x) + ki+j) = 0. (7)A question of convergence of the infinite sum is delicate and requires special considerationin each case.

Evident condition W∞(x) = W ′∞(x) = 0, which is still related to the solitondynamics, is necessary for rigorous justification of (7). Here we shall operate with formalseries and assume that initial chain (2) always may be recovered by adding (7) for i = n andi = n+1.

In order to find infinite number of superpotentials {Wi} from (7) one has to relatethem to one unknown function via some simple rule. Following Ref.

[3] we take the AnsatzWi(x) = qiW(qix),(8)which yields the equationW ′(x) + W 2(x) −γ2 + 2∞Xj=1(−1)jq2jW 2(qjx) = 0,(9)where γ2 = −P∞j=1(−1)jkj. Note that reality of superpotentials does not necessarily restrictparameter q to be real – this will appear later.

From (9) it is easy to derive eqs. (7) and (2)withki+1 = γ2(1 + q2)q2i,i ≥0.

(10)The following computationγ2 = −∞Xj=1(−1)jkj = γ2(1 + q2)∞Xj=0(−1)jq2j ≡γ2(11)shows that γ2 is completely arbitrary parameter (an energy scale) and (10) is a self-consistentdefinition of the constants ki.Derivation (11) is valid only at |q| < 1, which was the2

restriction of Ref. [3], but if (8) and (10) are taken as the basic substitutes for (2) then bydefinition γ2 is arbitrary and there are no essential restrictions on q up to now.Eq.

(9) has certain relation to quantum algebras [1] and corresponding q-analysis [10].In order to see this we first introduce a scaling operator Tq obeying group lawTqf(x) = f(qx),TqTp = Tqp,T −1q= Tq−1,T1 = 1. (12)Then (9) can be rewritten asW ′(x) −W 2(x)=γ2 −2∞Xj=0(−1)j(q2Tq)jW 2(x)=γ2 −2(1 + q2Tq)−1W 2(x).

(13)Multiplying (13) from the l.h.s. by (1+q2Tq) we obtain finite-difference-differential equationdefining W(x)W ′(x) + W 2(x) + qW ′(qx) −q2W 2(qx) = γ2(1 + q2),(14)which is nothing else than the first iteration of superpotentials.

The whole infinite chain (2)is thus generated by (14). This observation removes ambiguities arising in (9) due to theconvergence problems.Let us try to find quantum mechanical spectrum generated by the self-similar potentialU0(x) associated to (14).

Suppose that eigenfunctions (4) are normalizable. Then potentialUi+1(x) contains one eigenvalue less than Ui(x), i.e.

there should be the following orderingof levelsE0 < E1 < . .

. < E∞,En = 12nXi=0ki = −12γ2 1 + q21 −q2q2n,(15)where we chose undefined constant k0 to be k0 = −γ2(1 + q2)/(1 −q2).

At negative γ2 it isnot possible to fulfil the ordering and at positive γ2 the parameter q should be real and lie inone of the regions |q| < 1 or |q| > 1. Taking the normalization γ2 = ω2|1 −q2|/(1 + q2) anddenoting |q| = exp (±η/2), η > 0, we arrive at exponentially small or large bound energyspectrumEn = ∓12 ω2 e∓ηn.

(16)What type of potentials these spectra would correspond to? In order to know this oneshould solve equation (14).

Then everything crucially depends on the normalizability of ψ(0)0in (4) because all other wave functions ψ(n)0are related to it by scaling. Normalizability isinsured if W(x) is a continuous function positive at x →+∞and negative at x →−∞.Under such conditions W(x) has at least one zero and we choose corresponding point to bex = 0, i.e.

W(0) = 0. Eq.

(14) now automatically leads to W(−x) = −W(x) and belowwe restrict q to be semipositive. Let us find solution of (14) in the Taylor series form nearthe zero.

Substituting an expansion W(x) =P∞i=1 cix2i−1 into (9) we obtain the followingrecursion relation for the coefficients cici = q2i −1q2i + 112i −1i−1Xm=1ci−mcm,i ≥2,c1 = γ2,(17)3

which at q = 0, γ = 1 generates Bernoulli numbers B2i, ci = 22i(22i −1)B2i/(2i)!. One maysay that (17) defines q-analogs of the Bernoulli numbers [Bi]q.

Equation (17) works well forall values of q. At q < 1 it describes q-deformation of the hyperbolic tangent, since at q = 0one hasW ′ + W 2 = γ2,W(x) = γ tanh γx,(18)which is one-level (soliton) superpotential associated to the Rosen-Morse problem.

At q > 1one has q-deformation of the trigonometric tangent which is recovered in the limit q →∞,W ′ −W 2 = γ2,W(x) = γ tan γx. (19)This superpotential creates an infinite-level P¨oschl-Teller potential U1(x) with the restrictedregion of coordinate definition: −π < 2γx < π.

On this finite cut U0(x) = 0 presents aninfinitely deep potential well. If one sets γ = 0 simultaneously with q or q−1 then conformalsuperpotentials, W(x) = ±1/x, are emerging.

Finally, at q = 1 one gets a standard harmonicoscillator problem.If q ̸= 0, 1, ∞, there is no analytical expression for W(x) but some general propertiesof this function may be found along the analysis of Ref. [1], where it was proven that forq < 1 superpotential is positive at x = +∞.

In this case required normalizability conditionis fulfilled and relation (16) with upper signs really corresponds to physical spectrum.At q > 1 the radius of convergence of the series defining W(x) is finite, rc < ∞. Frominequalitiesγ2ω2 ≡q2 −1q2 + 1 < q2i −1q2i + 1 < 1,i > 1we have 0 < c(1)i< ci < c(2)i , where c(1,2)iare defined by the rule (17) when q-factor on ther.h.s.

is replaced by γ2/ω2 and 1 respectively (c(1,2)1= c1). As a result, 1 < 2γrc/π < ω/γ,which means that W(x) is smooth only on some cut at the ends of which it has singularities.From the basic relation (14) it follows that there is an infinite number of simple ”primary”and ”secondary” poles.

The former ones have residues equal to −1 and their location pointsxm tend to π(m + 1/2)/γ, m ∈Z, at q →∞.”Secondary” poles are situated at x =qnxm, n ∈Z+, and corresponding residues are defined by some algebraic equations. We arethus forced to consider Shr¨odinger operators (3) on a cut [−x1, x1] and impose boundaryconditions ψ(i)n (±x1) = 0 although the potential U0(x) is finite at x = ±x1.

The structureof W(x) leads to ψ(0)0 (±x1) = 0, i.e. ψ(0)0is true ground state of H0.

Note, however, thatthe spectrum En for such type of problems can not grow faster than n2 at n →∞inapparent contradiction with (16). This discrepancy is resolved by observation that alreadyW1(x) = qW(qx) has singularities on the interval [−x1, x1] so that only H0 and H1 areisospectral in the chain (3).

Hence, the positive signs case of (16) does not correspond toreal physical spectrum of the model.The number of deformations of a given function is not limited. The crucial propertypreserved by the presented above q-curling is the property of exact solvability of ”unde-formed” Rosen-Morse, harmonic oscillator, and P¨oschl-Teller potentials.

It is well knownthat potentials at infinitely small and exact zero values of a parameter may obey completelydifferent spectra. In our case, deformation with q < 1 converts one-level problem (18) with4

E0 = −γ2/2 into the infinite-level one with exponentially small energy eigenvalues (16).Whether one gets exactly solvable potential at q > 1 is an open question but this is quiteplausible because at q = ∞a problem with known spectrum En = γ2(n + 1)2/2 arises.In standard dynamical symmetry approach Hamiltonian of a system is supposed to beproportional either to Casimir operator or to polynomial combination of the generators ofsome Lie algebra [8, 11]. As a result, energy eigenvalues are determined by rational functionsof quantum numbers.

This means that one does not go out of the universal enveloping alge-bra. q-Deformation of the universal algebra works with functions (exponentials) of generatorsand, as it was announced, accounts for the presented exponential spectra.Indeed, substituting superpotentials (8) into relation (6) one finds raising and loweringoperatorsψ(0)n±1 ∝A±ψ(0)n ,H0 = 12(A+A−−1 + q21 −q2γ2),A+ = q1/2(p + iW(x))Tq,A−= q−1/2T −1q (p −iW(x)),(20)For real q and γ the operators A± are hermitian conjugates of each other.

Eq. (14) insuresthe following q-commutation relationsA−A+ −q2A+A−= γ2(1 + q2),H0A± = q±2A±H0.

(21)Introduction of formal number operatorN = ln H0/E0ln q2,N ψ(0)n= n ψ(0)n ,[N, A±] = ±A±(22)completes the definition of q-deformed harmonic oscillator algebra in the particular form [2].The quantum conformal algebra suq(1, 1) is realized as follows [12],K+ = (qγ(1 + q2)q−N/2A+)2,K−= (K+)†,K0 = 12(N + 12),[K0, K±] = ±K±,[K+, K−] = −q4K0 −q−4K0q2 −q−2,(23)i.e. it is a dynamical symmetry algebra of the model.

Generators K± are parity invariantand therefore even and odd wave functions belong to different irreducible representations ofsuq(1, 1).In order to generalize basic equation (14) we introduce an additional parameter s intothe superpotential, W = W(x, s), and assume that Tq in (20) is a generalized shift operatorTqW(x, s) = W(qx + a, s + 1),(24)where q and a are parameters of affine transformation.Although A+ is not hermitianconjugate of A−any more, we force them to obey q-oscillator type algebraA−A+ −q2A+A−= C(s),A±C(s) = C(s ± 1)A±,5

where C is some function of s. Resulting equation for the superpotentialW ′(x, s −1) + qW ′(qx + a, s) + W 2(x, s −1) −q2W 2(qx + a, s) = C(s),(25)may be called the generalized shape-invariance condition (cf. [9]).We define a Hamiltonian H as followsH = 12A+A−+ F(s),q2F(s) −F(s −1) = 12C(s),(26)where finite-difference equation for the function F(s) is found from the braiding relationsHA± = q±2A±H.

Now it is easy to generalize formula (15). Suppose that a wave functionψ0, A−ψ0 = 0, is normalizable.

Then a tower of higher states ψn ∝(A+)nψ0 gives energyspectrumEn = F(s) + 12nXi=1q2(i−1)C(s + i) = q2nF(s + n),(27)which can be found by purely algebraic means.If for some n = N normalizability ofψn is broken then H has only N discrete levels. In the above presentation we chose thesimplest form of s-parameter transformation under the action of Tq-operator.

One can easilygeneralize formula (27) for arbitrary change of variable s in (24), s →f(s).To conclude, in this paper we have described an exactly solvable quantum mechanicalproblem where quantum algebra suq(1, 1) acts on the discrete set of energy eigenstates scalingtheir eigenvalues by the constant factor. In the original version of differential geometricapplications of quantum Lie algebras an underlying space was taken to be non-commutative(”quantum plane”) and deformation parameter q was measuring deviations from normalanalysis (see, e.g., Ref.[13]).

Here we have commutative space and standard one-dimensionalquantum mechanics but the potential is very peculiar. It represents q-deformation of exactlysolvable potentials so that the spectrum remains to be known but it acquires essentiallyfunctional character.It is interesting to know the most general exactly solvable q-deformed potential.

Oneof the approaches to this problem consists in repetition of the trick described in Ref. [14].Namely, one can take as particle’s wave function a q-hypergeometric function multipliedby some weight factor.

This would correspond to the transformation of q-hypergeometricequation to the form of standard Schr¨odinger equation for some potential. Another path toq-deformation of known models is given by the eq.

(25) which may have solutions generalizingthose found by the old factorization technique at q = 1, a = 0.Two final remarks are in order. First, affine transformations appearing in (25) may beused for the definition of q-deformed supersymmetric quantum mechanics [15].

Second, atcomplex values of q one has meaningful dynamical systems which are exactly solvable whenq is a root of unity [16].The author is indebted to A.Shabat for acquainting with his paper prior to publicationand for relevant remarks. This work was supported by the NSERC of Canada.6

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[16] A.Shabat and V.Spiridonov, unpublished.7

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