Scattering theory of discrete (pseudo) Laplacians on a Weyl chamber

TL;DR

This paper develops a scattering theory for discrete pseudo Laplacians linked to crystallographic root systems, providing explicit wave and scattering operators, and applies it to hyperbolic Ruijsenaars-Schneider models related to Macdonald polynomials.

Contribution

It introduces a novel scattering framework for multivariate orthogonal polynomials associated with root systems and integrable lattice models, with explicit formulas for wave and scattering operators.

Findings

Explicit closed-form wave operators and scattering operators derived.

Application to hyperbolic Ruijsenaars-Schneider models elucidates scattering behavior.

Connects orthogonal polynomials, integrable systems, and scattering theory.

Abstract

To a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. As an application, we describe the scattering behavior of certain hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials.