Separation of variables in time-dependent Schrodinger equations

We classify (1 + 3)-dimensional Schrodinger equations for a particle interacting with an electromagnetic field that are solvable by the method of separation of variables into second-order ordinary differential equations. It is established, in particular, that the necessary condition for the Schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables. We describe vector-potentials that (a) provide the separability of the Schrodinger equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe a nonzero magnetic field. Furthermore, we apply the results obtained for separating variables to the Hamilton-Jacobi equation.