On the two-dimensional stationary Schrodinger equation with a singular potential

We study the equation -Delta u(x, y) + v(x, y)u(x, y) = 0 when the potential v has the following expression v(r, theta) = A(theta)/r(2) where (r, 0) are the polar coordinates in R(2) and A is a 2 pi-periodic function of class C(k) with k >= 1. We obtain series representations for solutions in a full neighborhood of the singular point and we also give representations in terms of pseudoanalytic formal powers in sectors having the singular point as a vertex. The results are obtained with the aid of the reduction of the stationary Schrodinger equation to a Vekua equation of a special form and by using recent developments in pseudoanalytic function theory. (C) 2010 Elsevier Inc. All rights reserved.