Institute of Mathematics of NAS of Ukraine,
3 Tereshchenkivska Street, 01601 Kyiv-4, UKRAINE
e-mail: zhaliy@imath.kiev.ua
http://arXiv.org/find/math/1/au:+zhalij/0/1/0/all/0/1

Separation of variables in Pauli equations.

Abstract:
Using our classification of separable Schroedinger equations with three space dimensions published in the paper [R. Zhdanov and A. Zhalij, J. Math. Phys., 40, No. 12, 6319 (1999)], we give an exhaustive description of the (1+3)-dimensional Pauli equations for a spin 1/2 particle interacting with the electro-magnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the vector-potentials of the electro-magnetic field providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable is that it is equivalent to the system of two Schroedinger equations and, furthermore, the magnetic field has to be independent of the spatial variables.
Moreover, we describe the expicit forms of the vector-potentials of the electro-magnetic field that (a) provide separability of Pauli equation and (b) satisfy the vacuum Maxwell equations without currents. Furthermore, we construct inequivalent coordinate systems enabling us to separate variables in the corresponding Pauli equation and carry out variable separation.
We prove, that solutions with separated variables of the Pauli equation are common eigenfunctions of three mutually commuting symmetry operators of Pauli equation. For a number of obtained potentials these operators are constructed in explicit form.