On the two-dimensional stationary Schrِdinger 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 expression ν ( r , θ ) = A ( θ ) r 2 where ( 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.