Common algebraic structure for the Calogero - Sutherland models

Abstract

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.One dimensional quantum integrable models with long-range interaction have attracted much interest, because of not only their physical significance, but also their beautiful mathematical structure. One of such models is the Sutherland (trigonometric) model, which describes interacting particles on a circle [1]. The total momentum and Hamiltonian of the model are respectively given by

A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero–Sutherland Type

“…These combinatorial results also include a formula resulting from the use of Sutherland's original solution algorithm [51]. We also mention operator solutions of the Calogero-and Sutherland models obtained in [5,24,25,52,53] and [32], respectively, as well as integral representations of the Jack polynomials; see [2,40,41] and references therein. Our results in this paper provide an explicit construction of the many-variable polynomials P n .…”

Computing with Differential-difference Operators

“…The commutative algebra generated by these operators has been used in the study of certain exactly solvable models of quantum mechanics, namely the Calogero-Sutherland-Moser models, which deal with systems of identical particles in a one-dimensional space (for example, see Hikami, 1996;Kakei, 1996;Lapointe and Vinet, 1996). Here we are concerned with the algebra A of operators on polynomials generated by…”

Nonlinear generalized Dunkl-wave equations and applications

“…Over the last years, much attention has been paid to these operators in various mathematical (and even physical) directions. In this prospect, Dunkl operators are naturally connected with certain Schrödinger operators for Calogero-Sutherland-type quantum many-body systems [1,7,12,13,15]. Moreover, Dunkl operators allow generalizations of several analytic structures, such as Laplace operator, Fourier transform, heat semigroup, wave equations, and Schrödinger equations [2,6,8,[20][21][22][23].…”