Baxter operators in Ruijsenaars hyperbolic system II. Bispectral wave functions

TL;DR

This paper demonstrates that wave functions in the quantum Ruijsenaars hyperbolic system diagonalize Baxter operators, establishing duality relations and bispectral properties, and introduces new integral representations.

Contribution

It proves the duality and bispectral properties of wave functions in the quantum Ruijsenaars system, extending previous results and confirming conjectured symmetries.

Findings

Wave functions diagonalize Baxter Q-operators.

Established duality relations for wave functions.

Derived new integral representations of wave functions.

Abstract

In the previous paper we introduced a commuting family of Baxter Q-operators for the quantum Ruijsenaars hyperbolic system. In the present work we show that the wave functions of the quantum system found by M. Halln\"as and S. Ruijsenaars also diagonalize Baxter operators. Using this property we prove the conjectured duality relation for the wave function. As a corollary, we show that the wave function solves bispectral problems for pairs of dual Macdonald and Baxter operators. Besides, we prove the conjectured symmetry of the wave function with respect to spectral variables and obtain new integral representation for it.