Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems

TL;DR

This paper derives kernel functions for quantum relativistic Toda systems, linking them to B"acklund transformations and their classical limits, expanding understanding of integrable systems.

Contribution

It introduces new kernel functions for relativistic Toda systems and connects them to B"acklund transformations, bridging quantum and classical integrable models.

Findings

Kernel functions for quantum relativistic Toda systems are obtained.

Limits of these kernel functions yield B"acklund transformations.

Results include nonrelativistic counterparts aligning with existing literature.

Abstract

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of B\"acklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.