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

Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials

The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same orthogonal polynomials. We show that this interesting property is inherited by the Ruijsenaars-Schneider-van Diejen (RSvD) systems, which are integrable deformation of the CSM systems; the equilibrium positions of the multi-particle classical RSvD systems and the eigenfunctions of the corresponding single-particle quantum RSvD systems are described by the same orthogonal polynomials, the continuous Hahn (special case), Wilson and Askey-Wilson polynomials. They belong to the Askey-scheme of the basic hypergeometric orthogonal polynomials and are deformation of the Hermite, Laguerre and Jacobi polynomials, respectively. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.Comment: 16 pages, 1 figur

Equilibrium Positions and Eigenfunctions of Shape Invariant (`Discrete') Quantum Mechanics

Certain aspects of the integrability/solvability of the Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen systems with rational and trigonometric potentials are reviewed. The equilibrium positions of classical multi-particle systems and the eigenfunctions of single-particle quantum mechanics are described by the same orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.Comment: 30 pages, 1 figure. Contribution to proceedings of RIMS workshop "Elliptic Integrable Systems" (RIMS, Nov. 2004

Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials

We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-P\"oschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.Comment: 13 page

Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators

We investigate the structure of the elliptic algebra U_{q,p}(^sl_2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs of the Drinfeld currents. They enable us to identify U_{q,p}(^sl_2) with the tensor product of U_q(^sl_2) and a Heisenberg algebra generated by P,Q with [Q,P]=1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element c. The vertex operators of Lukyanov and Pugai arise as `intertwiners' of U_{q,p}(^sl_2) for level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.Comment: 49 pages, (AMS-)LaTeX ; added an explanation of integration contours; added comments. To appear in Comm. Math. Phys. Numbering of equations is correcte