Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.Comment: 7 pages; one reference added, published in Phys. Lett. B682 (2009) 130-13

Equivalences of the Multi-Indexed Orthogonal Polynomials

Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters.Comment: 25 pages. Some comments and a reference added. To appear in J.Math.Phy

Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts

In our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the $q$-Racah polynomial and its reduced form polynomials are obtained.Comment: 37 pages. Comments, a reference and proportionality constants for q-Racah case are added. Sec.3.3 is moved to App.B. To appear in PTE

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III

In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by G\'omez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type.Comment: 37 pages. Comments added, typo in (A.15) corrected, reference information updated. To appear in J.Math.Phy

Shape Invariant Potentials in "Discrete Quantum Mechanics"

Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant ``discrete quantum mechanical systems" are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are "discrete" counterparts of Calogero and Sutherland systems, the celebrated exactly solvable multi-particle dynamics. Deformed Hermite and Laguerre polynomials are the typical examples of the eigenfunctions of the above shape invariant discrete quantum mechanical systems.Comment: 15 pages, 1 figure. Contribution to a special issue of Journal of Nonlinear Mathematical Physics in honour of Francesco Calogero on the occasion of his seventieth birthda

Extensions of solvable potentials with finitely many discrete eigenstates

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than n_{max} so that their energies are lower than the groundstate energy.Comment: 22 pages, 3 figures. Typo corrected, comments and references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1212.659

Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

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 Schr\"odinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.Comment: 46 pages, 2 figure

Comments on the Deformed W_N Algebra

We obtain an explicit expression for the defining relation of the deformed W_N algebra, DWA(^sl_N)_{q,t}. Using this expression we can show that, in the q-->1 limit, DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} reduces to the sl_N-version of the Lepowsky-Wilson's Z-algebra of level k, ZA(^sl_N)_k. In other words DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} can be considered as a q-deformation of ZA(^sl_N)_k. In the appendix given by H.Awata, S.Odake and J.Shiraishi, we present an interesting relation between DWA(^sl_N)_{q,t} and \zeta-function regularization.Comment: 10 pages, LaTeX2e with ws-ijmpb.cls, Talk at the APCTP-Nankai Joint Symposium on ``Lattice Statistics and Mathematical Physics'', 8-10 October 2001, Tianjin Chin