Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability

We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H$_2^+$ molecular ion. The symmetry operator is constructed explicitly. We give the details of the Hamiltonian reduction of the 3D system to a 2D system with modified potential that is separable in elliptic coordinates. The potentials for which there is double-periodicity of the Schr\"odinger operator in the space of prolate spheroidal coordinates, including one for the H$_2^+$ molecular ion, are indicated. We study possible potentials that admit exact-solvability is as well as all models known to us with the (quasi)-exact-solvability property for the separation equations. We find deep connections between second-order superintegrable and conformally superintegrable systems and these tractable problems. In particular we derive a general 4-parameter expression for a model potential that is always integrable and is conformally superintegrable for some parameter choices.