Hilbert space theory for relativistic dynamics with reflection. Special cases

TL;DR

This paper introduces a new class of Hilbert space eigenfunction transforms that diagonalize operators related to relativistic two-particle Calogero-Moser dynamics, revealing unique scattering features and solutions to Yang-Baxter equations.

Contribution

It develops a novel framework for analyzing relativistic hyperbolic Calogero-Moser systems using eigenfunction transforms and explores their scattering properties and algebraic relations.

Findings

New eigenfunction transforms for relativistic Calogero-Moser dynamics

Distinctive reflection and transmission amplitudes with function-theoretic features

Amplitudes satisfy Yang-Baxter equations, linking to integrable models

Abstract

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero-Moser dynamics. The scattering is described by reflection and transmission amplitudes $t$ and $r$ with function-theoretic features that are quite different from nonrelativistic amplitudes. The axiomatic Hilbert space analysis in the appendices is inspired by and applied to the attractive two-particle relativistic Calogero-Moser dynamics for a sequence of special couplings. Together with the scattering function $u$ of the repulsive case, this leads to a triple of amplitudes $u, t, r$ satisfying the Yang-Baxter equations.