Separable quantizations of Stäckel systems

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (defined on 2n$2n$ dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations–parametrized by n$n$ arbitrary functions–that turn this system into a quantum separable Stäckel system.