Geodesic flows of c-projectively equivalent metrics are quantum integrable

Abstract

Given two c-projectively equivalent metrics on a Kähler manifold, we show that 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 commuting operators lead to separation of variables for Schrödinger’s equation.