Ruijsenaars spectral transform

TL;DR

This paper introduces a spectral transform based on Ruijsenaars hyperbolic system wave functions, generalizing Fourier transforms, and establishes its inversion, orthogonality, and unitarity properties across different regimes.

Contribution

It provides the first rigorous proof of inversion and orthogonality relations for the Ruijsenaars spectral transform for complex parameters.

Findings

Inversion formula and orthogonality relations established

Unitarity regimes of the transform characterized

Transform defines isomorphisms of $L_2$ spaces in four regimes

Abstract

Spectral decomposition with respect to the wave functions of Ruijsenaars hyperbolic system defines an integral transform, which generalizes classical Fourier integral. For a certain class of analytical symmetric functions we prove inversion formula and orthogonality relations, valid for complex valued parameters of the system. Besides, we study four regimes of unitarity, when this transform defines isomorphisms of the corresponding $L_2$ spaces.