Integrable quantum St\"ackel systems

TL;DR

This paper classifies a specific class of Stäckel separable systems, showing that only Benenti systems retain quantum integrability and separability after quantization.

Contribution

It identifies that within monomial-entry Stäckel systems, only Benenti systems preserve quantum integrability and separability.

Findings

Benenti systems are the only class with commuting quantum integrals.

Benenti systems are uniquely quantum separable within the studied class.

Quantum integrability is preserved only for Benenti systems in this class.

Abstract

The St\"ackel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. In the present paper we consider a class of St\"ackel separable systems where the entries of the St\"ackel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the St\"ackel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems (that is, they admit separation of variables in the Schr\"odinger equation) in the class under study.