Sturm-Liouville operators with distributional potentials

Abstract

In this paper we propose four different methods to determine Sturm-Liouville operator on an interval $(a,b)$ in case, when a potential $q(x)$ is a distribution from the Sobolev space with negative index of smoothness, i.e. (q\in W_2^{-\theta}), where (\theta\le 1). The main and second terms of asymptotic series for eigenvalues and eigenfunctions of these operators are obtained and the remaining terms are estimated depending on the class of smoothness of potential /(q/). Particular families of potentials, not belonging to the space (W_2^{-1}) are studied as well.