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

Abstract

We study the equation − u(x, y) + ν(x, y)u(x, y) = 0 when the potential ν has the following expressionwhere (r, θ) are the polar coordinates in R 2 and A is a 2π -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 Schrödinger equation to a Vekua equation of a special form and by using recent developments in pseudoanalytic function theory.