Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

TL;DR

This paper explores the quantization of natural Hamiltonian systems with quadratic constants of motion, focusing on the geometric conditions for maintaining integrability through quantum corrections involving the Laplace-Beltrami operator.

Contribution

It provides a complete geometric characterization of quantum corrections needed for quantizing integrable systems with quadratic constants of motion.

Findings

Derived conditions for quantum corrections in natural Hamiltonian systems

Analyzed Stäckel systems in various geometric settings

Provided examples in conformally flat and non-flat manifolds

Abstract

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. St\"ackel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.