Geodesic flows of c-projectively equivalent metrics are quantum integrable

TL;DR

This paper proves that geodesic flows of c-projectively equivalent metrics on Kähler manifolds are quantum integrable, with commuting quantum operators derived from classical integrals, and extends results to natural Hamiltonian systems with potentials.

Contribution

It establishes the quantum integrability of geodesic flows for c-projectively equivalent metrics and generalizes the results to systems with potentials, providing a new link between classical and quantum integrability.

Findings

Classical integrals of motion become commuting quantum operators.

Quantum integrability holds for geodesic flows of c-projectively equivalent metrics.

Separation of variables is achieved for Schrödinger's equation in these systems.

Abstract

Given two c-projectively equivalent metrics on a K\"ahler manifold we show that the canoncially constructed, Poisson-commuting integrals of motion of the geodesic flow, linear and quadratic in momenta, also commute as quantum operators. The methods employed here also provide a proof of a similar statement in the case of projective equivalence. We also investigate the addition of potentials, i.e. the generalization to natural Hamiltonian systems. We show that the commuting operators lead to separation of variables for Schr\"odinger's equation.