Darboux Transformations of Bispectral Quantum Integrable Systems

TL;DR

This paper develops a higher-dimensional Darboux transformation framework for quantum integrable systems using bispectral duality, enabling the construction of new integrable models from known systems.

Contribution

It introduces a novel approach to Darboux transformations in quantum integrable systems based on bispectral properties, expanding the toolkit for generating new integrable models.

Findings

Constructed Darboux transformations for quantum systems using bispectral duality.

Generated new quantum integrable systems from trivial and known bispectral systems.

Applied the method to systems like quantum Calogero-Moser.

Abstract

We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero-Moser.