Baxter operators in Ruijsenaars hyperbolic system III. Orthogonality and completeness of wave functions

TL;DR

This paper proves the orthogonality and completeness of wave functions in the quantum Ruijsenaars hyperbolic system, establishing their unitarity and advancing the understanding of the system's spectral properties.

Contribution

It demonstrates orthogonality and completeness of wave functions using Baxter Q-operators and duality, confirming the unitarity of the associated integral transform.

Findings

Wave functions diagonalize Baxter Q-operators

Orthogonality relations are established

Completeness relations are proven

Abstract

In the previous paper we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform.