Exactly Solvable Discrete Quantum Mechanics; Shape Invariance, Heisenberg Solutions, Annihilation-Creation Operators and Coherent States

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1 663 Progress of Theoretical Physics, Vol. 9, No. 4, April 2008 Exactly Solvable Discrete Quantum Mechanics; Shape Invariance, Heisenberg Solutions, Annihilation-Creation Operators and Coherent States Satoru Odake and Ryu Sasaki 2 Department of Physics, Shinshu University, Matsumoto , Japan 2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto , Japan (Received February 2, 2008 Various examples of exactly solvable discrete quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions of the symmetry algebra of the Askey-Wilson system are explored in detail.. Introduction General theory of exactly solvable discrete quantum mechanics of one degree of freedom systems is presented with all known examples. The discrete quantum mechanics is a simple extension or ormation of quantum mechanics in which the momentum operator p appears in the Hamiltonian in the exponentiated forms e ±γp, γ R, instead of polynomials in ordinary quantum mechanics. The corresponding Schrödinger equations are difference equations with imaginary shifts, instead of differential. The eigenfunctions of the exactly solvable discrete quantum mechanics of one degree of freedom systems consist of the (q-askey-scheme of hypergeometric orthogonal polynomials,, 2 which are ormations of the classical orthogonal polynomials, like the Hermite, Laguerre, Jacobi polynomials, etc., 3 constituting the eigenfunctions of exactly solvable ordinary quantum mechanics. 4, 5 These eigenpolynomials are orthogonal with respect to absolutely continuous measure functions, which are just the square of the ground state wavefunctions; a familiar situation in quantum mechanics. For another type of orthogonal polynomials with discrete measures,, 2, 6 see Ref. 7 for a unified theory. Like most exactly solvable quantum mechanics, every example of exactly solvable discrete quantum mechanics is endowed with dynamical symmetry, shape invariance, 8 which allows to determine the entire energy spectrum and the corresponding eigenfunctions when combined with Crum s theorem 9 or the factorisation method. 4, 5 In other words, shape invariance guarantees exact solvability in the Schrödinger picture. 0 2 As expected, exact solvability in the Heisenberg picture also holds for all these examples. The explicit forms of Heisenberg operator solutions give rise to the explicit expressions of annihilation/creation operators as the positive/negative frequency parts. 3 The annihilation/creation operators together with the Hamiltonian constitute the dy-

2 664 S. Odake and R. Sasaki namical symmetry algebra. In some cases, the algebras are simple and tangible, like the oscillator algebra and its q-ormations, 4 or su(,. The present paper is to supplement or to complete some results in previous publications. 0 3 The discrete quantum mechanics of the Meixner-Pollaczek, the continuous Hahn, the continuous dual Hahn, the Wilson and the Askey-Wilson polynomials discussed in Refs. 0 3 are only for restricted parameter ranges; for example the angle was φ = π/2 for the Meixner-Pollaczek polynomial and all the parameters were restricted real for the continuous Hahn, the continuous dual Hahn, the Wilson and the Askey-Wilson polynomials. This is due to a historical reason that these polynomials with the restricted parameter ranges were first recognised by the present authors as describing the classical equilibrium positions 0 2, 5 7 of multi-particle exactly solvable dynamical systems of Ruijsenaars-Schneider-van Diejen type. 8, 9 It is a ormation of the classical results dating as far back as Stieltjes, that the classical equilibrium positions of multi-particle exactly solvable dynamical systems of Calogero-Sutherland type 23, 24 are described by the zeros of the classical orthogonal polynomials (the Hermite, Laguerre and Jacobi. The discrete quantum mechanics was constructed 0 2 based on the analogy that these orthogonal polynomials would constitute the eigenfunctions of certain quantum mechanical systems in the same way as the classical orthogonal polynomials (the Hermite, Laguerre and Jacobi do. As will be shown in detail in the main text, these orthogonal polynomials enjoy the exact solvability and related properties for the full ranges of the parameters. Attempts to further orm these exactly solvable systems have yielded several examples of the so-called quasi-exactly solvable systems. 28, 29 Another objective of the present paper is to explore in detail the properties of the systems obtained by restricting the Askey-Wilson system, treated in Some of these have interesting and useful forms of the dynamical symmetry algebras or the explicit forms of coherent state, etc., as evidenced by the q-oscillator algebras realised by the continuous (big q-hermite polynomial. 4 Aspects of ordinary theory of orthogonal polynomials are not particularly emphasised. This paper is organised as follows. In 2, the general setting of the discrete quantum mechanics is recapitulated with appropriate notation. Starting with the parameters in the potential function and the Hamiltonian, various concepts and solution methods are briefly surveyed. Sections 3 to 5 are the main body of the paper, discussing various examples of exactly solvable discrete quantum mechanics. They are divided into three groups according to the sinusoidal coordinate η(x. Section 3 is for the polynomials in η(x =x. Section 4 is for the polynomials in η(x =x 2. Section 5 is for the polynomials in η(x =cosx. Very roughly speaking, polynomials in 3 are the ormation of the Hermite polynomial; those in 4 aretheormation of the Laguerre polynomial and those in 5 are the ormation of the Jacobi polynomial from the point of view of the sinusoidal coordinates, but not from the energy spectrum. Section 6 is for a summary and comments. Appendix A provides a diagrammatic proof of the hermiticity (self-adjointness of the Hamiltonians of discrete quantum mechanics. Appendix B is a collection of the inition of basic symbols and functions used in this paper for self-containedness.

3 Exactly Solvable Discrete Quantum Mechanics General setting The dynamical variables are the coordinate x (x R and the conjugate momentum p, which is realised as a differential operator p = id/dx. The other parameters are symbolically denoted as λ =(λ,λ 2,... on top of q (0 <q< and φ (φ R. For the q-systems, the parameters are denoted as q =(q λ,q λ 2,... Complex conjugation is denoted by and the absolute value f(x is f(x = f(xf(x.here f(x means (f(x and f(x x x+a = f(x + a,sincex is real. Hamiltonian The Hamiltonian has a general form H = V (x e γp V (x + V (x e γp V (x V (x V (x, (2. in which γ is a real constant. It is either or log q. The potential function V depends on the parameters, V (x =V (x ; λ, whereas the q and φ dependence is not explicitly indicated. The parameter dependence of the Hamiltonian H = H(λ is not explicitly indicated in most cases. The eigenvalue problem or the time-independent Schrödinger equation is a difference equation instead of differential in ordinary quantum mechanics: Hφ n (x =E n φ n (x (n =0,, 2,..., E 0 < E < E 2 <, (2.2 in which φ n (x =φ n (x ; λ is the eigenfunction belonging to the energy eigenvalue E n = E n (λ. The difference equation has inherent non-uniqueness of solutions; if φ(x is a solution so is φ(xq(x when Q(x is any periodic function with the period iγ. This non-uniqueness problem is resolved when the Hilbert space of the state vectors is specified. See Appendix A. Factorisation Factorisation of the Hamiltonian is an important property H = T + + T V (x V (x =(S + S (S + S =A A, (2.3 in which various quantities S ± = S ± (λ, T ± = T ± (λ, A = A(λ are ined as ( denotes the hermitian conjugation with respect to the chosen inner product (2.75 and (A. (A.3: S + = e γp/2 V (x, S = e γp/2 V (x, S + = V (x e γp/2, S = V (x e γp/2, (2.4 T + = S + S + = V (x e γp V (x, T = S S = V (x e γp V (x, (2.5 A = i(s + S, A = i(s + S. (2.6 Ground state wavefunction The ground state wavefunction φ 0 (x =φ 0 (x ; λ is annihilated by the A operator Aφ 0 (x =0 Hφ 0 (x =0 E 0 =0, (2.7

4 666 S. Odake and R. Sasaki which is a zero mode of the Hamiltonian. The above equation reads explicitly as V (x + iγ2 φ 0 (x iγ2 = V (x + iγ 2 φ 0(x + iγ 2. (2.8 Among possible solutions, we choose a real and nodeless φ 0. As will be shown in Appendix A, the requirement of the hermiticity (self-adjointness of the Hamiltonian H selects a unique solution φ 0, which is given explicitly in each subsection (3.0, (3.25, (4.7, (4.23, (5., (5.38, (5.59, (5.79, (5.97, (5.27 and (5.46. Similarity transformed Hamiltonian The similarity transformed Hamiltonian H = H(λ in terms of the ground state wavefunction φ 0 (2.8 is in which T ± are ined as H = φ 0 (x H φ 0 (x = T + + T V (x V (x = V (x e γp + V (x e γp V (x V (x, (2.9 T + = φ 0 (x T + φ 0 (x =V (x e γp, T = φ 0 (x T φ 0 (x =V (x e γp. (2.0 It acts on the polynomial part of the eigenfunction. Let us write the excited state eigenfunction φ n (x =φ n (x ; λ as φ n (x ; λ =φ 0 (x ; λp n (η(x;λ, (2. in which P n (η =P n (η ; λ is a polynomial in the sinusoidal coordinate η(x. 3 Here η(x is a real function of x. The sinusoidal coordinate η(x discussedinthispaper has no λ-dependence in contrast to the cases studied in Ref. 7. Then H acts on P n (η: H(λP n (η(x;λ =E n (λp n (η(x;λ. (2.2 For all the examples discussed in this paper, H is lower triangular in the special basis, η(x, η(x 2,...,η(x n,..., (2.3 spanned by the sinusoidal coordinate η(x (η(x =x, x 2, cos x: 0 3 H(λη(x n = E n (λη(x n +lowerordersinη(x. (2.4 Shape invariance The factorised Hamiltonian (2.3 has the dynamical symmetry called shape invariance 8 if the following relation holds: A(λA(λ = κa(λ + δ A(λ + δ+e (λ, (2.5 in which κ is a real positive parameter and δ denotes the shift of the parameters and E (λ is the eigenvalue of the first excited state. This relation is satisfied by all the examples discussed in this paper. Shape invariance means that the original Hamiltonian H(λ and the associated Hamiltonian A(λA(λ in Crum s 9 sense (or the susy partner Hamiltonian in the so-called supersymmetric quantum mechanics 4, 5

5 Exactly Solvable Discrete Quantum Mechanics 667 have the same shape up to a multiplicative factor κ and an additive constant E (λ. In terms of the potential function V (x ; λ, the above relation reads explicitly as V (x iγ 2 ; λv (x + iγ 2 ; λ = κ 2 V (x ; λ + δv (x + iγ ; λ + δ, (2.6 V (x + iγ 2 ; λ+v (x + iγ 2 ; λ = κ ( V (x ; λ + δ+v (x ; λ + δ E (λ. (2.7 Among many consequences of shape invariance, we list the most salient ones. All the eigenvalues are generated by E (λ and the corresponding eigenfunctions are generated from the known form of the ground state eigenfunction φ 0 (2.7 together with the multiple action of the successive A operator: 0 2 n E n (λ = κ s E (λ + sδ, (2.8 s=0 φ n (x ; λ A(λ A(λ + δ A(λ +2δ A(λ +(n δ φ 0 (x ; λ + nδ. (2.9 The latter is related to a Rodrigues type formula for the eigenpolynomials. We illustrate the shape invariance and Crum s scheme in Fig. at the end of this section. The Hilbert space belonging to the Hamiltonian H(λ isdenotedash. Closure relation Another important symmetry concept of exactly solvable quantum mechanics is the closure relation: 7, 3 [H, [H,η]]=ηR 0 (H+[H,η] R (H+R (H. (2.20 Here η(x is the sinusoidal coordinate and R i (H is a polynomial in H. At the classical mechanics level, it is easy to see that the closure relation means that η(x undergoes a sinusoidal motion with frequency R 0 (E. The closure relation (2.20 is satisfied by all the examples discussed in this paper and the explicit forms of R i (H, i =, 0, ande n (λ are given in each subsection. The closure relation (2.20 enables us to express any multiple commutator [H, [H,, [H,η(x] ]] as a linear combination of the operators η(x and [H,η(x] with coefficients depending on the Hamiltonian H only. As we will see shortly, the exact Heisenberg operator solution and the annihilation/creation operators are obtained as a consequence. 7, 3 Let us consider the closure relation (2.20 as an algebraic constraint on η(x and the Hamiltonian, for given constants {r (j i }. The l.h.s. consists of e 2γp, e γp,,e γp, e 2γp, then R i can be parametrised as R 0 (y =r (2 0 y2 + r ( 0 y + r(0 0, R (y =r ( y + r(0, R (y =r (2 y2 + r ( y + r(0. (2.2 The similarity transformation of (2.20 [ H, [ H,η]]=ηR 0 ( H+[ H,η] R ( H+R ( H (2.22 gives rise to the following five conditions: η(x 2iγ 2η(x iγ+η(x =r (2 0 η(x+r(2 + ( r( η(x iγ η(x, (2.23 η(x +2iγ 2η(x + iγ+η(x =r (2 0 η(x+r(2 + ( r( η(x + iγ η(x, (2.24

6 668 S. Odake and R. Sasaki ( ( η(x iγ η(x V (x iγ+v (x + iγ V (x V (x = ( r (2 ( 0 η(x+r(2 V (x iγ+v (x + iγ + V (x+v(x r ( ( ( η(x iγ η(x V (x iγ+v (x + iγ + r ( 0 η(x+r( + ( r(0 η(x iγ η(x, (2.25 ( ( η(x + iγ η(x V (x iγ + V (x + iγ V (x V (x = ( r (2 ( 0 η(x+r(2 V (x iγ + V (x + iγ+v (x + V (x r ( ( ( η(x + iγ η(x V (x iγ + V (x + iγ + r ( 0 η(x+r( + ( r(0 η(x + iγ η(x, ( ( η(x η(x iγ V (xv (x + iγ +2 ( η(x η(x + iγ V (x V (x + iγ = ( r (2 ( 0 η(x+r(2 V (xv (x + iγ + V (x V (x + iγ+ ( V (x+v(x 2 + r ( ( η(x iγ η(x V (xv (x + iγ + r ( ( η(x + iγ η(x V (x V (x + iγ ( r ( ( 0 η(x+r( V (x+v(x + r (0 0 η(x+r(0. (2.27 For real {r (j i } (this is indeed the case for all the examples discussed in this paper, (2.24 and (2.26 are the complex conjugate of (2.23 and (2.25, respectively. In contrast to the cases of the orthogonal polynomials with discrete measures discussed in 4 of Ref. 7, the determination of η(x and the possible forms of V (x is not straightforward due to the ambiguities of periodic functions with iγ period. Here we mention only the basic results. It is easy to see that (2.23 (2.26 require r (2 0 = r ( and r ( 0 =2r (0, which is consistent with the hermitian conjugation of (2.20. With these constraints, the first condition (2.23 reads with x x + iγ η(x iγ (2 + r ( η(x+η(x + iγ =r(2. (2.28 Following the arguments given in 4 and Appendix A of Ref. 7, we deduce from (2.25 and (2.27 the general relationship ( ( η(x iγ η(x η(x + iγ η(x (V (x+v(x = r (0 η(x2 r ( η(x C (x, (2.29 ( ( η(x 2iγ η(x η(x iγ η(x + iγ V (xv (x + iγ ( (0 r η(x iγη(x+r( = η(x iγ+c (x ( r (0 η(x iγη(x+r( η(x+c (x ( 2 η(x iγ η(x r (0 0 η(x iγη(x r(0 ( η(x iγ+η(x + C2 (x, (2.30 in which C j (x (j =, 2 is an arbitrary function satisfying the periodicity C j (x + iγ =C j (x. The hermiticity of the Hamiltonian H would restrict C j (x severely. Further analysis of the closure relation (2.23 (2.27 will be published elsewhere. Like the cases of discrete measures, 7 the dual closure relation [η, [η, H]]=H R dual 0 (η+[η, H] R dual (η+r dual (η (2.3

7 Exactly Solvable Discrete Quantum Mechanics 669 holds and Ri dual are given by R dual (η(x = ( η(x iγ η(x + ( η(x + iγ η(x, (2.32 R0 dual (η(x = ( η(x iγ η(x ( η(x + iγ η(x, (2.33 R dual (η(x = ( V (x+v(x R0 dual (η(x. (2.34 Equations (2.28 and (2.29 imply R dual (y =r ( y+r(2 r ( η(x+c (x. and Rdual (η(x = r(0 η(x2 + Auxiliary function ϕ In all the examples discussed in this paper, the ground state wavefunction with shifted x and parameters φ 0 (x iγ 2 ; λ + δ is related to its original value φ 0 (x ; λ via a real auxiliary function ϕ: φ 0 (x iγ 2 ; λ + δ = V (x ; λ ϕ(x iγ 2 φ 0(x ; λ. (2.35 The auxiliary function ϕ(x discussed in this paper has no λ-dependence in contrast to the cases studied in Ref. 7. It is easy to see that (2.35 implies (2.8. The explicit forms of ϕ(x are given at the beginning of each section (3., (4., (5.. Similarity transformation II Similarity transformed Hamiltonian or that of S ±, S ± operators (2.4 take simpler forms with the help of the auxiliary function ϕ (2.35: φ 0 (x ; λ + δ S ± (λ φ 0 (x ; λ =ϕ(x e ±γp/2, (2.36 { φ 0 (x ; λ S ± (λ V (x ; λ e γp/2 ϕ(x, φ 0 (x ; λ + δ = V (x ; λ e γp/2 (2.37 ϕ(x. Note that the parameter shifts ±δ are properly incorporated. Forward/Backward shift operators With (2.36 (2.37 the similarity transformed A and A operators are obtained. They are called the forward/backward shift operators: H(λ =B(λF(λ, (2.38 F(λ = φ 0 (x ; λ + δ A(λ φ 0 (x ; λ =iϕ(x ( e γp/2 e γp/2, (2.39 B(λ = φ 0 (x ; λ A(λ φ 0 (x ; λ + δ = i ( V (x ; λ e γp/2 V (x ; λ e γp/2 ϕ(x. (2.40 The action of the forward shift operator F(λ and the backward shift operator B(λ on the polynomial P n (η ; λ are: F(λP n (η ; λ =f n (λp n (η ; λ + δ, (2.4 B(λP n (η ; λ + δ =b n (λp n+ (η ; λ, (2.42 in which f n (λ andb n (λ are real constants related to E n (λ: f n (λb n (λ =E n (λ. (2.43

8 670 S. Odake and R. Sasaki For the cases studied in Ref. 7 b n (λ is actually independent of n, but here it depends on n. In terms of the forward and backward shift operators, the shape invariance condition (2.5 reads F(λB(λ =κb(λ + δf(λ + δ+e (λ. (2.44 Corresponding to (2.9, a Rodrigues type formula for the eigenpolynomials is P n (η ; λ = B(λ b n (λ B(λ + δ b n 2 (λ + δ B(λ +2δ B(λ +(n δ b n 3 (λ +2δ b 0 (λ +(n δ P 0(η ; λ + nδ, (2.45 where P 0 (η ; λ + nδ = for all the examples given in this paper. With these quantities the action of A(λ anda(λ on the eigenfunction φ n can be simply expressed as A(λφ n (x ; λ =f n (λφ n (x ; λ + δ, (2.46 A(λ φ n (x ; λ + δ =b n (λφ n+ (x ; λ. (2.47 Three term recurrence relation The polynomial part of the eigenfunction P n (η is an orthogonal polynomial with the measure φ 0 (x 2. It satisfies three term recurrence relations., 2 Let us first write the relation for the monic polynomial Pn monic (η =η n + lower degree in η: P n (η =c n Pn monic (η, (2.48 Pn+ monic (η (η a rec n Pn monic (η+ n Pn monic (η =0 (n 0, (2.49 with P monic (η =0. ForP n (η itreads ηp n (η =A n P n+ (η+b n P n (η+c n P n (η, (2.50 A n = c n, B n = a rec n, C n = c n n. c n+ c n (2.5 Sometimes we write the parameter dependence explicitly as P n (η =P n (η ; λ, a rec n = a rec n (λ, n = n (λ, c n = c n (λ, A n = A n (λ, B n = B n (λ, C n = C n (λ, f n (λ and b n (λ. They are given in each subsection. Heisenberg operator and Annihilation and Creation operators The exact Heisenberg operator solution for η(x is easily obtained 3 from the closure relation (2.20: e ith η(xe ith = a (+ e iα +(Ht + a ( e iα (Ht R (HR 0 (H, (2.52 α ± (H = 2( R (H ± R (H 2 +4R 0 (H, (2.53 R (H =α + (H+α (H, R 0 (H = α + (Hα (H, (2.54 ( a (± = ± [H,η(x] ( η(x+r (HR 0 (H (α+ α (H (H α (H (2.55 = ± ( α + (H α (H ( [H,η(x] + α ± (H ( η(x+r (HR 0 (H. (2.56

9 Exactly Solvable Discrete Quantum Mechanics 67 The positive/negative frequency parts of the Heisenberg operator solution, a (± are the annihilation and creation operators a (+ = a (, a (+ φ n (x =A n φ n+ (x, a ( φ n (x =C n φ n (x. (2.57 Since α ± (E n =E n± E n, (2.58 we obtain a (± φ n (x = ± ( [H,η(x] + (E n E n η(x+ R (E n φ n (x. (2.59 E n+ E n E n± E n Commutation relations of a (± and H Simple commutation relations [H,a (± ]=a (± α ± (H (2.60 follow from (2.55 and (2.20. When applied to φ n, we obtain with the help of (2.58, [H,a (± ]φ n =(E n± E n a (± φ n. (2.6 Commutation relations of a (± are expressed in terms of the coefficients of the three term recurrence relation by (2.57: a ( a (+ φ n = A n C n+ φ n = n+φ n, a (+ a ( φ n = C n A n φ n = n φ n, (2.62 [a (,a (+ ]φ n =( n+ n φ n. (2.63 These relations simply mean the operator relations a ( a (+ = f(h, (2.64 a (+ a ( = g(h, (2.65 in which f and g are analytic functions of H. In other words, H and a (± form a so-called quasi-linear algebra. 30 This is because the inition of the annihilation/creation operators depend only on the closure relation (2.20, without any other inputs. The situation is quite different from those of the wide variety of proposed annihilation/creation operators for various quantum systems, 3 most of which were introduced within the framework of algebraic theory of coherent states. In all these cases there is no guarantee for symmetry relations like (2.64 and (2.65. In many cases it is convenient to introduce the number operator (or the level operator N N φ n = nφ n. (2.66 For the following types of energy spectra, the number operator N can be expressed as a function of the Hamiltonian H: E n = an (a >0 N = a H, (2.67 E n = n(n + b (b>0 N = H + 4 b2 2 b, (2.68 E n = q n q N =(H +, (2.69 E n =(q n ( bq n (0<b< q N = 2b( H + b + (H + b + 2 4b. (2.70

10 672 S. Odake and R. Sasaki Obviously the Hamiltonian is expressed as H = E N. Then (2.63 can be expressed simply as [a (,a (+ ]= N + N (2.7 and (2.6 is rewritten as With a ormed commutator [H,a (± ]=E N a (± a (± E N = a (± (E N± E N. (2.72 [A, B] α = AB αba, (2.73 we have [a (,a (+ ] α = N + α N. (2.74 Orthogonality and normalisation The scalar product for the elements of the Hilbert space belonging to the Hamiltonian H is (g, f = dx g(x f(x, (2.75 in which the integration range depends on the specific Hamiltonian or the polynomial. The orthogonality of the eigenvectors {φ n (x}, φ n (x =φ 0 (xp n (η(x is: (φ n,φ m = dx φ 0 (x ; λ 2 P n (η(x;λ P m (η(x;λ =h n (λδ nm, (2.76 in which h n (λ > 0. The constants h n, c n and n n = c2 n c 2 n h n h n (n, h n = h 0 c 2 n Letusdenotethen-th normalised eigenfunction as are related as n j= n (n 0. (2.77 ˆφ n (x ; λ =N n (λp n (η(x;λ ˆφ 0 (x ; λ, ˆφ0 (x ; λ = φ 0(x ; λ h0 (λ, N n(λ = h 0 (λ h n (λ. (2.78 These normalisation constants are given for each polynomial. Coherent states There are many different and nonequivalent initions of coherent states. Here we adopt the most conventional one, as the eigenvector of the annihilation operator a (,(2.57: a ( ψ(α, x =αψ(α, x, α C. (2.79 It is expressed in terms of the coefficient C n of the three term recurrence relation (2.50 and (2.5 as 3 ψ(α, x =ψ(α, x ; λ =φ 0 (x ; λ n=0 α n n k= C P n (η(x;λ. (2.80 k

11 Exactly Solvable Discrete Quantum Mechanics 673 Thus we obtain one new coherent state for each polynomial; (3.9, (3.39, (4.6, (4.37, (5.20, (5.5, (5.7, (5.9, (5.8, (5.37 and (5.58. If the sum on the r.h.s. is expressed by a simple function, it is a generating function of the polynomial P n (η. In most explicit examples to be discussed in later sections, the potential functions, the Hamiltonians and thus the polynomials themselves are totally symmetric in the parameters, see for example, the Askey-Wilson polynomial 5.. The above coherent state, being totally symmetric, gives the best candidate for a symmetric generating function. For the polynomials to be discussed in later sections, however, most of the known generating functions are not totally symmetric. λ-shift operators Let us fix an orthonormal basis { ˆφ n (x; λ} and ine a unitary operator U (U as U ˆφ n (x ; λ = ˆφ n (x ; λ + δ, U ˆφn (x ; λ + δ = ˆφ n (x ; λ. (2.8 Then we can ine another set of annihilation-creation operators â, â : â = U A, â = A U. (2.82 They satisfy H =â â and their actions on φ n are derived from (2.46 and (2.47, âφ n (x ; λ φ n (x ; λ, â φ n (x ; λ φ n+ (x ; λ. Although this kind of creation and annihilation operators have been considered in many literature, 3 it should be stressed that they are formal because U and U are formal operators. On the other hand, a (± obtained from the Heisenberg solution are explicitly expressed in terms of difference operators (differential operators, in ordinary quantum mechanics, (2.55. Note that the construction method of â and â is based on the shape invariance but that of a (± is not. The latter is based on the closure relation. The key point of the construction of â and â is the proper shift of the parameters λ, which is achieved by the formal operators U and U. We introduce another set of λ-shift operators X and X explicitly in terms of difference operators through the following relations: a (+ = A X, a ( = X A. (2.83 By using the shape invariance (2.5, we have Aa (+ = AA X = ( κa(λ + δ A(λ + δ+e X = ( κh(λ + δ+e X. (2.84 Since κh(λ + δ+e is a positive operator, we obtain X = ( κh(λ + δ+e A a (+ = ( A κh(λ + δ+e ( [H,η(x] ( η(x+r (HR 0 (H (α+ α (H (H α (H. (2.85 Similarly X is expressed as X = a ( A ( κh(λ + δ+e. (2.86

12 674 S. Odake and R. Sasaki Fig.. Shape invariance and Crum s scheme. Their actions on φ n are Xφ n (x ; λ = A n(λ b n (λ φ n(x ; λ + δ, (2.87 X φ n (x ; λ + δ = C n+(λ f n+ (λ φ n(x ; λ, (2.88 and the λ-shift without changing the level n is achieved, as expected. The λ-shift operators for the polynomials P n (η(x;λ aregivenbyφ 0 (x ; λ + δ X φ 0 (x ; λ and φ 0 (x ; λ X φ 0 (x ; λ + δ. The expression of X and X may be simplified for some particular cases (see 3.2, 4.2 and 5.5. Finally we illustrate the shape invariance and Crum s scheme in Fig.. The Hilbert space belonging to the Hamiltonian H(λ is denoted as H. The action of various operators and their domains and images are also illustrated in Fig.: H(λ, a (± (λ, â(λ, â(λ : H H, (2.89 A(λ, X(λ, U(λ : H H +, (2.90 A(λ,X(λ, U(λ : H + H. (2.9

13 Exactly Solvable Discrete Quantum Mechanics η(x =x From this section to 5, we present various formulas and results specific to each example of the exactly solvable discrete quantum mechanics. These examples are divided into three groups according to the form of the sinusoidal coordinate; η(x =x in this section, η(x =x 2 in 4, η(x =cosx in 5. The names of the subsections are taken from the name of the corresponding orthogonal polynomial and the number, for example, [KS.4] indicates the corresponding subsection of the review of Koekoek and Swarttouw. 6 In all the examples in this section, we have η(x =x, <x<, γ =, κ =, ϕ(x =. ( continuous Hahn [KS.4] In previous works, 0 3 the parameters a and a 2 were restricted to real, positive values. Now they are complex with positive real parts. parameters and potential functions λ =(a,a 2, δ =( 2, 2 ; Re a i > 0; V (x ; λ =(a + ix(a 2 + ix. (3.2 shape invariance and closure relation E n (λ =n(n + b, (3.3 R (y =2, R 0 (y =4y + b (b 2, (3.4 R (y = i(a + a 2 a 3 a 4 y i(b 2(a a 2 a 3 a 4, (3.5 4 b = a j, (a 3,a 4 =(a,a 2 or (a 2,a. (3.6 j= These can be rewritten as eigenfunctions E n (λ =n(n +2Re(a + a 2, (3.7 R 0 (y =4y +4Re(a + a 2 ( Re(a + a 2, (3.8 R (y =2Im(a + a 2 y +4 ( Re(a + a 2 Im(a a 2. (3.9 φ 0 (x ; λ = Γ (a + ixγ (a 2 + ix = Γ (a + ixγ (a 2 + ixγ (a 3 ixγ (a 4 ix, (3.0 P n (η ; λ =p n (x ; a,a 2,a 3,a 4 = i n (a + a 3 n (a + a 4 ( n n, n + a + a 2 + a 3 + a 4, a + ix 3F 2, n! a + a 3,a + a 4 (3.

14 676 S. Odake and R. Sasaki which are symmetric under a a 2 and a 3 a 4 separately. c n = (n + b n, (3.2 ( n! = i a (n + b (n + a + a 3 (n + a + a 4 (2n + b (2n + b, (3.3 a rec n + n(n + a 2 + a 3 (n + a 2 + a 4 (2n + b 2(2n + b n = n(n + b j= k=3 (n + a j + a k (2n + b 3(2n + b 2 2 (2n + b, (3.4 f n (λ =n + b, b n (λ =n +. (3.5 annihilation/creation operators and commutation relations α ± (H =± 2 H, H = H + 4 (b 2, (3.6 N = H 2 (b (for b >, (3.7 [H,a (± ]=a (± ( ± 2 H. (3.8 The annihilation/creation operators (2.55 and their commutation relation (2.63 are not so simplified because n+ brec n = (quartic polynomial in n/(cubic polynomial in n has a lengthy expression. coherent state (b 2n α n ψ(α, x ; λ =φ 0 (x ; λ 2 4 j= k=3 (a P n (η(x;λ. (3.9 j + a k n n=0 The r.h.s is symmetric under a a 2 and a 3 a 4 separately. We are not aware if a concise summation formula exists or not. Several non-symmetric generating functions for the continuous Hahn polynomial are given in Ref. 6. orthogonality 2 4 φ 0 (x ; λ 2 j= k=3 P n (η ; λp m (η ; λdx =2π Γ (n + a j + a k n!(2n + b Γ (n + b δ nm, (3.20 h 0 (λ = Γ (b 2π 2 4 j= k=3 Γ (a j + a k, h 0 (λ h n (λ = b +2n n!(b n b + n 2 4 j= k=3 (a. j + a k n ( Meixner-Pollaczek [KS.7] In previous works, 0, 3, 32 the parameter φ was fixed to π/2. Here we treat the most general case 0 <φ<π. parameters and potential function λ = a, δ = 2, φ (0 <φ<π; a>0; V (x ; λ = e i( π 2 φ (a + ix. (3.22

15 Exactly Solvable Discrete Quantum Mechanics 677 shape invariance and closure relation E n (λ =2nsin φ, (3.23 R (y =0, R 0 (y =4sin 2 φ, R (y =2y cos φ +2asin 2φ. (3.24 eigenfunctions φ 0 (x ; λ = e (φ π 2 x Γ (a + ix, (3.25 e 2iφ, (3.26 P n (η ; λ =P n (a (x ; φ = (2a n n! c n = (2 sin φn n! a rec n ( n, a + ix e inφ 2F 2a = n + a tan φ, brec n = n(n +2a (2 sin φ 2, (3.27 f n (λ =2sinφ, b n (λ =n +. (3.28 The polynomial has the following symmetry P n (a (x ; φ =P n (a ( x ; φ. annihilation/creation operators and commutation relations α ± (H =±2sinφ, N = 2sinφ H, (3.29 a (± = ± 4sinφ [H,η]+ 2 η + cos φ 4sin 2 φ (H +2asin φ, (3.30 n+ n = n + a 2sin 2 φ, (3.3 [H,a (± ]=±2sinφa (±, (3.32 [a (,a (+ ]= 4sin 3 φ (H +2asin φ. (3.33 su(, algebra : J ± =2sinφa (±, J 3 = (H +2asin φ, 2sinφ [J 3,J ± ]=±J ±, [J,J + ]=2J 3. (3.34 The su(, or sl(2, R algebra reported before 3, 32 is a special case of the present one. λ-shift operators For the special case of φ = π/2 the annihilation/creation operators are closely related to the A and A operators: a (+ = A X, X = 4 (S + + S, (3.35 a ( = X A, X = 4 (S + + S, (3.36 φ 0 (x ; λ + δ X(λ φ 0 (x ; λ P n (η ; λ = 2 P n(η ; λ + δ, (3.37 φ 0 (x ; λ X(λ φ 0 (x ; λ + δ P n (η ; λ + δ = 4 (n +2aP n(η ; λ. (3.38

16 678 S. Odake and R. Sasaki coherent state The coherent state gives a simple generating function, which generalises the previous result: 3 orthogonality ψ(α, x ; λ =φ 0 (x ; λ (2 sin φ n α n P n (η(x;λ (2a n n=0 = φ 0 (x ; λ e iα( e2iφ F ( a + ix 2a φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π (2 sin φ2a = h 0 (λ 2πΓ(2a, 4iα sin 2 φ. (3.39 Γ (n +2a n!(2sinφ 2a δ nm, (3.40 h 0 (λ h n (λ = n!. (3.4 (2a n The exact solvability of the continuous Hahn and Meixner-Pollaczek polynomials for the full parameters are discussed in Ref. 27 in connection with their further ormation to give another example of quasi exactly solvable system. 4. η(x =x 2 In all the examples in this section, we have η(x =x 2, 0 <x<, γ =, κ =, ϕ(x =2x. ( Wilson [KS.] The Wilson polynomial is the most general one in this category. The parameters a,..., a 4 were restricted to real positive values in previous works. 0 3 The generic situation to be discussed in this paper is {a,a 2,a 3,a 4} = {a,a 2,a 3,a 4 } (as a set, Re a i > 0 ( i 4. (4.2 parameters and potential function λ =(a,a 2,a 3,a 4, δ =( 2, 2, 2, 2 ; V (x; λ = (a + ix(a 2 + ix(a 3 + ix(a 4 + ix. 2ix(2ix + (4.3 shape invariance and closure relation E n (λ =n(n + b, (4.4 R (y =2, R 0 (y =4y + b (b 2, R (y = 2y 2 +(b 2b 2 y +(2 b b 3, (4.5 4 b = a j, b 2 = a j a k, b 3 = a j a k a l. (4.6 j= j<k 4 j<k<l 4

17 Exactly Solvable Discrete Quantum Mechanics 679 eigenfunctions φ 0 (x ; λ 4 = j= Γ (a j + ix Γ (2ix, (4.7 P n (η ; λ =W n (x 2 ; a,a 2,a 3,a 4 =(a + a 2 n (a + a 3 n (a + a 4 n ( n, n + 4 j= 4 F a j, a + ix, a ix 3, (4.8 a + a 2,a + a 3,a + a 4 which are symmetric under the permutations of (a,a 2,a 3,a 4. c n =( n (n + b n, (4.9 a rec n = (n + b 4 j=2 (n + a + a j + n 2 j<k 4 (n + a j + a k a 2 (2n + b (2n + b (2n + b 2(2n + b, (4.0 n = n(n + b 2 j<k 4 (n + a j + a k (2n + b 3(2n + b 2 2 (2n + b, (4. f n (λ = n(n + b, b n (λ =. (4.2 annihilation/creation operators and commutation relations α ± (H =± 2 H, H = H + 4 (b 2, (4.3 N = H 2 (b (for b >, (4.4 [H,a (± ]=a (± ( ± 2 H. (4.5 The annihilation/creation operators (2.55 and their commutation relation (2.63 are not so simplified because the expression n+ brec n = (a degree 0 polynomial in n/ (a degree 7 polynomial in n is quite complicated. coherent state ( n (b 2n α n ψ(α, x ; λ =φ 0 (x ; λ n! j<k 4 (a P n (η(x;λ. (4.6 j + a k n n=0 The r.h.s. is symmetric under the permutations of (a,a 2,a 3,a 4. It is not known to us if a concise summation formula exists or not. Several non-symmetric generating functions for the Wilson polynomial are given in Ref. 6. orthogonality φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx 0 j<k 4 =2πn!(n + b Γ (n + a j + a k n δ nm, (4.7 Γ (2n + b h 0 (λ = Γ (b 2π j<k 4 Γ (a j + a k, h 0 (λ h n (λ = b +2n (b n b + n n! j<k 4 (a. j + a k n (4.8

18 680 S. Odake and R. Sasaki 4.2. continuous dual Hahn [KS.3] This is a restricted case of the Wilson polynomial with a 4 =0. Inprevious works, 0 3 the parameters a, a 2 and a 3 were real and positive. Now they are {a,a 2,a 3 } = {a,a 2,a 3 },asasetandrea i > 0. This is dual to the continuous Hahn 3. in the sense that the roles of η(x ande n are interchanged. For the continuous Hahn, η(x =x and E n is quadratic in n, whereasη(x isquadraticin x and E n = n for the dual Hahn. The duality has sharper meaning for polynomials with discrete orthogonality measures, see for example Ref. 7. parameters and potential function λ =(a,a 2,a 3, δ =( 2, 2, 2 ; V (x ; λ = (a + ix(a 2 + ix(a 3 + ix. (4.9 2ix(2ix + shape invariance and closure relation E n (λ =n, (4.20 R (y =0, R 0 (y =, R (y = 2y 2 +( 2b y b 2, (4.2 b = a + a 2 + a 3, b 2 = a a 2 + a a 3 + a 2 a 3. (4.22 eigenfunctions φ 0 (x ; λ 3 = j= Γ (a j + ix Γ (2ix, (4.23 P n (η ; λ =S n (x 2 ; a,a 2,a 3 ( n, a + ix, a ix =(a + a 2 n (a + a 3 n 3 F 2, (4.24 a + a 2,a + a 3 which are symmetric under the permutations of (a,a 2,a 3. c n =( n, a rec (4.25 n =(n + a + a 2 (n + a + a 3 +n(n + a 2 + a 3 a 2, (4.26 = n (n + a j + a k, (4.27 n j<k 3 f n (λ = n, b n (λ =. (4.28 annihilation/creation operators and commutation relations α ± (H =±, N = H, (4.29 a (± = ± 2 [H,η]+ 2 η H2 (b 2 H 2 b 2, (4.30 n+ n =4n 3 +3(2b n 2 + ( 2b (b + 2b 2 + n + b b 2 a a 2 a 3. (4.3

19 Exactly Solvable Discrete Quantum Mechanics 68 The interesting algebra, reported in Ref. 3, with H 3 non-linearity on the r.h.s. of (4.33 is valid for the full parameter range: [H,a (± ]=±a (±, (4.32 [a (,a (+ ]=4H 3 +3(2b H 2 + ( 2b (b + 2b 2 + H + b b 2 a a 2 a 3. (4.33 λ-shift operators ( 3 X = is + T + + x iv (x i 2 j= i (2a j 8( + x is T + (x + iv (x i 2 +i j= (2a j 8( + x 2 S + S, (4.34 φ 0 (x ; λ + δ X(λ φ 0 (x ; λ P n (η ; λ =P n (η ; λ + δ, (4.35 φ 0 (x ; λ X(λ φ 0 (x ; λ + δ P n (η ; λ + δ = (n + a j + a k P n (η ; λ. (4.36 coherent state ψ(α, x ; λ =φ 0 (x ; λ n=0 j<k 3 ( n α n n! j<k 3 (a j + a k n P n (η(x;λ. (4.37 The r.h.s. is symmetric under the permutations of (a,a 2,a 3. We are not aware if a concise summation formula exists or not. Several non-symmetric generating functions for the continuous dual Hahn polynomial are given in Ref. 6. orthogonality 0 φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2πn! h 0 (λ = 2π j<k 3 Γ (a j + a k, j<k 3 Γ (n + a j + a k δ nm, (4.38 h 0 (λ h n (λ = n! j<k 3 (a. (4.39 j + a k n 5. η(x =cosx In all the examples in this section, we have η(x =cosx, 0 <x<π, γ=logq, κ = q, ϕ(x =2sinx. (5. Throughout this paper q is always in the range 0 <q< and this will not be indicated. It is convenient to introduce a complex variable z = e ix. Then the shift operator e γp canbewrittenas e γp = e iγ d dx We have changed the sign of ϕ(x from Ref. 3. = q z d dz, (5.2

20 682 S. Odake and R. Sasaki whose action on a function of x can be expressed as z qz: e γp f(x =f(x iγ =q z d dz ˇf(z = ˇf(qz, with f(x = ˇf(z. Note that γ< Askey-Wilson [KS3.] The Askey-Wilson polynomial is the most general one with the maximal number of parameters, four. All the other polynomials in this section are obtained by restricting the parameters a,...,a 4, in one way or another. In previous publications 0 3 these restricted polynomials were not discussed individually, since their exact solvability is a simple corollary of that of the Askey-Wilson. However, the simpler structure of the restricted ones would give rise to simple energy spectrum and interesting and tractable forms of the dynamical symmetry algebras and coherent states, etc., as exemplified by the continuous q-hermite polynomial 5.5, which has a = a 2 = a 3 = a 4 = 0. It gives a most natural realisation of the q-oscillator algebra. 4 parameters and potential function q =(a,a 2,a 3,a 4, δ =( 2, 2, 2, 2, q; (5.3 V (x ; λ = ( a z( a 2 z( a 3 z( a 4 z ( z 2 ( qz 2, z = e ix. (5.4 The parameters have to satisfy the conditions {a,a 2,a 3,a 4} = {a,a 2,a 3,a 4 } (as a set, a i <, i =,...,4. (5.5 In previous works 0 3 only the real parameters a i R were discussed. shape invariance and closure relation E n (λ =(q n ( b 4 q n, (5.6 R (y =(q 2 q 2 2 y, y = y ++q b 4, (5.7 R 0 (y =(q 2 q 2 2 ( y 2 ( + q 2 b 4, (5.8 R (y = 2 (q 2 q 2 2 ( (b + q b 3 y ( + q (b 3 + q b b 4, (5.9 4 b = a j, b 3 = a j a k a l, b 4 = a a 2 a 3 a 4. (5.0 j= j<k<l 4 eigenfunctions φ 0 (x ; λ = (e 2ix ; q 4 j= (a, (5. je ix ; q P n (η ; λ =p n (cos x ; a,a 2,a 3,a 4 q ( q = a n n (a,a a 2 a 3 a 4 q n,a e ix,a e ix q a 2,a a 3,a a 4 ; q n 4 φ 3 ; q, a a 2,a a 3,a a 4 (5.2

21 Exactly Solvable Discrete Quantum Mechanics 683 which are symmetric under the permutations of (a,a 2,a 3,a 4. c n =2 n (b 4 q n ; q n, (5.3 a rec n = ( a + a ( b 4q n 4 j=2 ( a a j q n 2 a ( b 4 q 2n ( b 4 q 2n a ( q n 2 j<k 4 ( a ja k q n ( b 4 q 2n 2 ( b 4 q 2n, (5.4 n = ( qn ( b 4 q n 2 j<k 4 ( a ja k q n 4( b 4 q 2n 3 ( b 4 q 2n 2 2 ( b 4 q 2n, (5.5 f n (λ =q n 2 (q n ( b 4 q n, b n (λ =q n+ 2. (5.6 annihilation/creation operators and commutation relations α ± (H = 2 (q 2 q 2 2 H ± 2 (q q H 2 4q b 4, H = H ++q b 4, (5.7 q N = q ( H H 2b 2 4q b 4 (for 0 <b 4 <q, (5.8 4 [H,a (± ]= 2 a(±( (q 2 q 2 2 H ± (q q H 2 4q b 4. (5.9 The annihilation/creation operators (2.55 and their commutation relation (2.63 are not simplified at all. The expression n+ brec n = q n (a degree 2 polynomial in q n /(a degree 6 polynomial in q 2n isverycomplicated. coherent state ψ(α, x ; λ =φ 0 (x ; λ n=0 2 n (b 4 ; q 2n α n (q ; q n j<k 4 (a ja k ; q n P n (η(x;λ. (5.20 The r.h.s. is symmetric under the permutations of (a,a 2,a 3,a 4. We are not aware if a concise summation formula exists or not. Several non-symmetric generating functions for the Askey-Wilson polynomial are given in Ref. 6. orthogonality π 0 φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π h 0 (λ = (q ; q j<k 4 (a ja k ; q, 2π(b 4 ; q (b 4 q n ; q n (b 4 q 2n ; q (q n+ ; q j<k 4 (a ja k q n ; q δ nm, (5.2 h 0 (λ h n (λ = b 4q 2n b 4 q n (b 4 ; q n (q ; q n j<k 4 (a ja k ; q n. (5.22

22 684 S. Odake and R. Sasaki 5... Askey-Wilson Wilson The Wilson polynomial is obtained from the Askey-Wilson polynomial by a q limit. Here we present a dictionary of the correspondence for future reference. Let us first introduce a new coordinate x for the Wilson polynomial as the rescaled one of the variable x (0 <x<π of the Askey-Wilson polynomial: x = L ( π x, 0 <x <L, p = π, L p γ = π L, λ =(a,a 2,a 3,a 4, (5.23 in which L is related to q as q = e π/l. This entails e γp = e p (5.24 and the following limit formulas as L or q : (The superscript W denotes the corresponding quantity for the Wilson polynomial. V (x ; λ lim L ( q 2 = V W (x ; λ, (5.25 lim L H(λ ( q 2 = HW (λ, lim L lim (q ; L q3 ( q 3 lim L P E n (λ ( q 2 = EW n (λ, (5.26 j a jφ 0 (x ; λ =φ W 0 (x ; λ, ϕ(x lim L q = ϕw (x, (5.27 P n (η(x;λ ( q 3n = P W n (η W (x ;λ, (5.28 lim ( qf(λ = F W (λ, L lim L lim L f n (λ ( q 2 = f W n (λ, (5.29 B(λ ( q 3 = BW (λ, lim L b n(λ = b W n (λ. ( continuous dual q-hahn [KS3.3] The continuous dual q-hahn polynomial is obtained by restricting a 4 =0in the Askey-Wilson polynomial 5.. This restriction renders the energy spectrum to a simple form E n = q n for all the restricted polynomials in 5 except for the continuous q-jacobi polynomial 5.6 and the continuous q-hahn polynomial For these the commutation relations of H and a (± is the same (5.47, (5.66, (5.86, (5.04 and (5.53. They can be expressed as q-ormed commutators (5.49, (5.68, (5.88, (5.06 and (5.55. The commutation relation [a (,a (+ ] or its ormation becomes drastically simpler, as the number of parameters decreases. parameters and potential function q =(a,a 2,a 3, δ =( 2, 2, 2, q; (5.3 V (x ; λ = ( a z( a 2 z( a 3 z ( z 2 ( qz 2, z = e ix. (5.32

23 Exactly Solvable Discrete Quantum Mechanics 685 The parameters have to satisfy the conditions {a,a 2,a 3} = {a,a 2,a 3 } (as a set, a i <, i =, 2, 3. (5.33 shape invariance and closure relation E n (λ =q n, (5.34 R (y =(q 2 q 2 2 y, y = y +, R 0 (y =(q 2 q 2 2 y 2, (5.35 R (y = 2 (q 2 q 2 2 ( (b + q b 3 y ( + q b 3, (5.36 b = a + a 2 + a 3, b 2 = a a 2 + a a 3 + a 2 a 3, b 3 = a a 2 a 3. (5.37 eigenfunctions φ 0 (x ; λ = (e 2ix ; q 3 j= (a, (5.38 je ix ; q P n (η ; λ =p n (cos x ; a,a 2,a 3 q ( q = a n (a,a e ix,a e ix a 2,a a 3 ; q n 3 φ 2 a a 2,a a 3 q ; q, (5.39 which are symmetric under the permutations of (a,a 2,a 3. c n =2 n, (5.40 a rec n = ( 2 a + a a ( a a 2 q n ( a a 3 q n a ( q n ( a 2 a 3 q n, (5.4 n = 4 ( qn ( a j a k q n, (5.42 j<k 3 f n (λ =q n 2 (q n, b n (λ =q n+ 2. (5.43 annihilation/creation operators and commutation relations α ± (H =(q (H +, q N =(H +, (5.44 a (± = ± ( q [H,η] q q ± +( q ± ( η 2 (b + q b ( + q b 3 (H + (H +, (5.45 n+ n = 4 (q 4 qb 2 3q 4n + 4 (q 3 b 3 (b 3 + qb q 3n 4 (q 2 (b b 3 + qb 2 q 2n + 4 (q (b 2 + qq n, (5.46 [H,a (± ]=(q a (± (H +, (5.47 [a (,a (+ ]= 4 (q 4 qb 2 3(H (q 3 b 3 (b 3 + qb (H (q 2 (b b 3 + qb 2 (H (q (b 2 + q(h +. (5.48

24 686 S. Odake and R. Sasaki The r.h.s. of the above commutation relation is a quartic polynomial in q N.Interms of a ormed commutator we have: Ha (± q a (± H =(q a (±, namely, [H,a (± ] q =(q a (±. (5.49 The following relation: n+ q 4 n = 4 ( qb 3(b 3 + qb q 3n + 4 ( q2 (b b 3 + qb 2 q 2n 4 ( q3 (b 2 + qq n + 4 ( q4 (5.50 meansthat[a (,a (+ ] q 4 is a cubic polynomial in q N. coherent state ψ(α, x ; λ =φ 0 (x ; λ orthogonality π 0 2 n α n (q ; q n j<k 3 (a P n (η(x;λ. (5.5 ja k ; q n n=0 φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π (q n+ ; q j<k 3 (a ja k q n δ nm, ; q h 0 (λ = 2π (q ; q j<k 3 (a j a k ; q, (5.52 h 0 (λ h n (λ = (q ; q n j<k 3 (a. ja k ; q n ( Al-Salam-Chihara [KS3.8] This is a further restriction of the continuous dual q-hahn polynomial 5.2 with a 3 = 0. The dynamical symmetry algebra is further simplified and [a (,a (+ ]isa quadratic polynomial in q N. The coherent state gives an explicit generating function with symmetry a a 2 (5.7. parameters and potential function q =(a,a 2, δ =( 2, 2,q; {a,a 2} = {a,a 2 } (as a set, a i <, i =, 2; (5.54 V (x ; λ = ( a z( a 2 z ( z 2 ( qz 2, z = eix. (5.55 shape invariance and closure relation E n (λ =q n, (5.56 R (y =(q 2 q 2 2 y, y = y +, R 0 (y =(q 2 q 2 2 y 2, (5.57 R (y = 2 (q 2 q 2 2 (a + a 2 y. (5.58

25 Exactly Solvable Discrete Quantum Mechanics 687 eigenfunctions φ 0 (x ; λ = (e 2ix ; q 2 j= (a, (5.59 je ix ; q P n (η ; λ =Q n (cos x ; a,a 2 q = a n (a a 2 ; q n 3 φ 2 ( q n,a e ix,a e ix a a 2, 0 which are symmetric under the permutations of (a,a 2. q ; q, (5.60 c n =2 n, a rec n = 2 (a + a 2 q n, n = 4 ( qn ( a a 2 q n, (5.6 f n (λ =q n 2 (q n, b n (λ =q n+ 2. (5.62 annihilation/creation operators and commutation relations α ± (H =(q (H +, q N =(H +, (5.63 a (± = ± ( q [H,η] q q ± +( q ± ( η 2 (a + a 2 (H +, (5.64 n+ n = 4 (q ( ( + qa a 2 q 2n +(a a 2 + qq n, (5.65 [H,a (± ]=(q a (± (H +, (5.66 [a (,a (+ ]= 4 (q ( ( + qa a 2 (H + 2 +(a a 2 + q(h +. (5.67 The r.h.s. is a quadratic polynomial in q N. The ormed commutators are: Ha (± q a (± H =(q a (±, namely, [H,a (± ] q =(q a (±. (5.68 Other interesting quantities are: n+ q n = 4 ( q( a a 2 q 2n, (5.69 n+ q 2 n = 4 ( q( +q (a a 2 + qq n. (5.70 These mean that [a (,a (+ ] q and [a (,a (+ ] q 2 take simple forms and, in particular, the latter is linear in q N. As we will see in another example, the continuous q- Laguerre 5.7, these are special to the restricted Askey-Wilson polynomials with a quadratic polynomial ( a z( a 2 z in the numerator of the potential function V (x (see (5.56 (5.57. coherent state ψ(α, x ; λ =φ 0 (x ; λ = φ 0 (x ; λ n=0 2 n α n (q, a a 2 ; q n P n (η(x;λ ( a e ix,a 2 e ix (2αe ix 2φ q ;2αe ix, (5.7 ; q a a 2 which is obviously symmetric in a a 2 and listed in Ref. 6.

26 688 S. Odake and R. Sasaki orthogonality π φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π 0 (q n+,a a 2 q n δ nm, (5.72 ; q h 0 (λ = 2π (q, a h 0 (λ a 2 ; q, h n (λ =. (5.73 (q ; q n (a a 2 ; q n 5.4. continuous big q-hermite [KS3.8] This is a further restriction of the Al-Salam-Chihara polynomial 5.3 with a 2 =0. The continuous big q-hermite gives another simple realisation of the q-oscillator algebra (5.90. parameters and potential function = a, δ = 2, q; <a<; (5.74 V (x ; λ az = ( z 2 ( qz 2, z = eix. (5.75 q shape invariance and closure relation E n (λ =q n, (5.76 R (y =(q 2 q 2 2 y, y = y +, R 0 (y =(q 2 q 2 2 y 2, (5.77 R (y = 2 (q 2 q 2 2 ay. (5.78 eigenfunctions φ 0 (x ; λ = (e 2ix ; q (ae ix ; q, (5.79 q ; q, (5.80 P n (η ; λ =H n (cos x ; a q = a n 3φ 2 ( q n,ae ix,ae ix 0, 0 c n =2 n, a rec n = 2 aqn, n = 4 ( qn, (5.8 f n (λ =q n 2 (q n, b n (λ =q n+ 2. (5.82 annihilation/creation operators and commutation relations α ± (H =(q (H +, q N =(H +, (5.83 a (± = ± ([H,η] q q q ± +( q ± (η 2 a (H +, (5.84 n+ n = 4 ( qqn, (5.85 [H,a (± ]=(q a (± (H +, (5.86 [a (,a (+ ]= 4 ( q(h +. (5.87 The ormed commutator makes (5.86 simpler Ha (± q a (± H =(q a (±, namely, [H,a (± ] q =(q a (±. (5.88

27 Exactly Solvable Discrete Quantum Mechanics 689 The relation n+ q n = 4 ( q, (5.89 implies another realisation of the q-oscillator a ( a (+ qa (+ a ( = 4 ( q, namely, [a(,a (+ ] q = 4 ( q. (5.90 coherent state (2.80 reads with the help of [KS(3.8.3] ψ(α, x ; λ =φ 0 (x ; λ orthogonality π 0 n=0 2 n α n (2αa ; q P n (η(x;λ =φ 0 (x ; λ (q ; q n (2αe ix, 2αe ix. ; q φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π h 0 (λ = 2π (q ; q, (5.9 (q n+ δ nm, ; q (5.92 h 0 (λ h n (λ =. (q ; q n ( continuous q-hermite [KS3.26] The continuous q-hermite polynomial has been discussed in some detail in Ref. 4 as the simplest dynamical system realising the q-oscillator algebra in two different ways (5.08 and (5.7. Here we recapitulate some formulas to make this paper complete. Like the Hermite polynomial, the continuous q-hermite has no parameter other than q. parameters and potential function V (x ; λ = shape invariance and closure relation ( z 2 ( qz 2, z = eix. (5.94 E n (λ =q n, (5.95 R (y =(q 2 q 2 2 y, y = y +, R 0 (y =(q 2 q 2 2 y 2, R (y =0. (5.96 eigenfunctions φ 0 (x ; λ = (e 2ix ; q, (5.97 q ; q n e 2ix, (5.98 ( P n (η ; λ =H n (cos x q q = e inx n, 0 2φ 0 c n =2 n, a rec n =0, n = 4 ( qn, (5.99 f n (λ =q n 2 (q n, b n (λ =q n+ 2. (5.00

28 690 S. Odake and R. Sasaki annihilation/creation operators and commutation relations α ± (H =(q (H +, q N =(H +, (5.0 a (± = ± ( q [H,η] q q ± +( q ± η (H +, (5.02 n+ n = 4 ( qqn, (5.03 [H,a (± ]=(q a (± (H +, (5.04 [a (,a (+ ]= 4 ( q(h +. (5.05 The formula (5.04 can be simplified as a ormed commutator: Ha (± q a (± H =(q a (±, namely, [H,a (± ] q =(q a (±. (5.06 The following relation: means a q-oscillator algebra: n+ q n = 4 ( q, (5.07 a ( a (+ qa (+ a ( = 4 ( q, namely, [a(,a (+ ] q = 4 ( q. (5.08 λ-shift operators Since the theory has no parameter λ, A and A work as the creation and annihilation operators. Thus a (+ and A (a ( and A are closely related: a (+ = A X, (5.09 A = i ( V (x e γp/2 V (x e γp/2, (5.0 X = i 2 q( z V (x e γp/2 z V (x e γp/2 (H +. (5. The similarity transformed quantities are: ã (+ = à X, ( à = φ 0 (x A φ 0 (x =q z 2 z 2 eγp/2 + ( X = φ 0 (x X φ 0 (x = 2 q 2 z 2 eγp/2 + (5.2 z z 2 e γp/2, (5.3 z 2 e γp/2 ( H +. (5.4 As there is no λ to be shifted, we have à = B(λ (2.40 and à = F(λ (2.39. The X and à operators work as XP n (x = 2 q n+ 2 Pn (x, à P n (x =q n+ 2 Pn+ (x (5.5 and X satisfies the relation ( 2q 2 ( 2 X( H + = z 2 eγp/2 + z 2 e γp/2 2 = V (xe γp + V (x e γp V (x V (x + = H +. (5.6

29 Exactly Solvable Discrete Quantum Mechanics 69 It is easy to verify that the shape invariance relation (2.5 itself implies a realisation of the q-oscillator algebra with A and A : 4 AA q A A = q, namely, [A, A ] q = q. (5.7 coherent state (2.80 reads with the help of [KS(3.26.] ψ(α, x ; λ =φ 0 (x ; λ orthogonality π 0 n=0 2 n α n P n (η(x;λ =φ 0 (x ; λ (q ; q n (2αe ix, 2αe ix. ; q φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π h 0 (λ = 2π (q ; q, 5.6. continuous q-jacobi [KS3.0] parameters and potential function (5.8 (q n+ δ nm, ; q (5.9 h 0 (λ h n (λ =. (q ; q n (5.20 λ =(α, β, δ =(,, q; α, β 2 ; (5.2 V (x ; λ = ( q 2 (α+ 2 z( q 2 (α+ 3 2 z( + q 2 (β+ 2 z( + q 2 (β+ 3 2 z ( z 2 ( qz 2, z = e ix. (5.22 shape invariance and closure relation E n (λ =(q n ( q n+α+β+, (5.23 R (y =(q 2 q 2 2 y, y = y ++q α+β+, (5.24 R 0 (y =(q 2 q 2 2 ( y 2 ( + q 2 q α+β, (5.25 R (y = 2 (q 2 q 2 2 q 4 ( + q 2 (q 2 α q 2 β ( q 2 (α+β ( y +(+qq (α+β 2. (5.26 eigenfunctions φ 0 (x ; λ = (e 2ix ; q (q 2 (α+ 2 e ix, q 2 (β+ 2 e ix ; q 2, (5.27 P n (η ; λ =P n (α,β (cos x q (5.28 = (qα+ ; q ( n q n,q n+α+β+,q 2 (α+ 2 e ix,q 2 (α+ 2 e ix 4φ 3 q ; q (q ; q n q α+, q 2 (α+β+, q 2 (α+β+2,

30 692 S. Odake and R. Sasaki 2 n q 2 (α+ 2 n (q n+α+β+ ; q n c n =, (5.29 (q, q 2 (α+β+, q 2 (α+β+2 ; q n a rec n = ( q 2 (α+ 2 + q 2 (α+ 2 2 ( qn+α+ ( q n+α+β+ ( + q n+ 2 (α+β+ ( + q n+ 2 (α+β+2 q 2 (α+ 2 ( q 2n+α+β+ ( q 2n+α+β+2 q 2 (α+ 2 ( q n ( q n+β ( + q n+ 2 (α+β ( + q n+ 2 (α+β+ ( q 2n+α+β ( q 2n+α+β+, (5.30 n =( q n ( q n+α ( q n+β ( q n+α+β ( + qn+ 2 (α+β ( + q n+ 2 (α+β 2 ( + q n+ 2 (α+β+ 4( q 2n+α+β ( q 2n+α+β 2 ( q 2n+α+β+, (5.3 f n (λ = q 2 (α+ 3 2 q n ( q n+α+β+ ( + q 2 (α+β+ ( + q 2 (α+β+2, (5.32 b n (λ =q 2 (α+ 3 2 q n+ (q (n+ ( + q 2 (α+β+ ( + q 2 (α+β+2. (5.33 annihilation/creation operators and commutation relations α ± (H = 2 (q 2 q 2 2 H ± 2 (q q H 2 4q α+β+, H = H ++q α+β+, (5.34 q N = 2 q α β ( H H 2 4q α+β+ (for 0 <q α+β+ <, (5.35 [H,a (± ]= 2 a(±( (q 2 q 2 2 H ± (q q H 2 4q. α+β+ (5.36 The annihilation/creation operators (2.55 and their commutation relation (2.63 are not so simplified because n+ brec n = q n (a degree 9 polynomial in q n / (a degree polynomial in q n has a lengthy expression. coherent state We are not aware if a simple summation formula exists for the coherent state: ψ(α (2q 2 (α+ 2 n (q 2 (α+β+ ; q 2 2n α n,x; λ =φ 0 (x ; λ (q α+,q β+ P n (η(x;λ. (5.37 ; q n n=0 orthogonality π 0 φ 0 (x ; λ 2 P n (η ; λp m (η ; λdx =2π ( qα+β+ (q α+,q β+, q 2 (α+β+3 ; q n q (α+ ( q 2n+α+β+ (q, q α+β+, q 2 (α+β+ ; q n (q 2 (α+β+2,q 2 (α+β+3 ; q (q, q α+,q β+, q 2 (α+β+, q 2 (α+β+2 ; q δ nm, ( n