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

We study the equation−△u(x,y)+ν(x,y)u(x,y)=0 when the potential ν has the following expressionν(r,θ)=A(θ)r2 where (r,θ) are the polar coordinates in R2 and A is a 2π-periodic function of class Ck 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.