208 research outputs found

### 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

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