On the new approach to variable separation in the two-dimensional Schrödinger equation

with some specific V (x1, x2) (see, e.g. [1–3] and references therein). Saying about the problem of SV in the Eq. (1), we imply two mutually connected problems. The first one is to describe all functions V (x1, x2) such that the equation (1) admits separation of variables (classification problem). The second problem is to construct for each function V (x1, x2) all coordinate systems making it possible to separate corresponding Schrödinger equation. As far as we know, the first problem has been solved provided V = 0 [3] and V = αx−2 1 + βx −2 2 [1] and the second one has not been considered in the literature at all. We guess that a possible reason for this was absence of an adequate mathematical technique to handle the classification problem. In the paper [4] we suggested a new approach to SV in partial differential equations which enabled us to solve the problem of SV in two-dimensional wave equation with time independent potential [4]. In the present paper we give the complete solution of the problem of SV in the Schrödinger equation (1) obtained within the framework of the above said approach. Solution with separated variables is looked for in the form of the ansatz [4]