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.Comment: 18 page
