Separation of variables in (1+3)-dimensional Schrodinger equations with vector-potential

Alexander Zhalij (alexzh@bgumail.bgu.ac.il)
Institute of Mathematics of the National Academy of Sciences of Ukraine
3 Tereshchenkivska Street
01601 Kyiv-4 UKRAINE

Department for Energy and Environmental Physics
The Jacob Blaustein Institute for Desert Research
Ben-Gurion University of the Negev Sede Boqer Campus
84 990, Israel

Abstract

We classify (1+3)-dimensional Schrodinger equations for a particle interacting with the 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 Schrodinger equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field.