Regularization of singular Sturm-Liouville equations

Abstract

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization corresponding operators are correctly defined as quasi-differential. Their resolvent approximation is investigated and all self-adjoint and maximal dissipative extensions and generalized resolvents are described in terms of homogeneous boundary conditions of the canonic form. Some results are new for the case $p(t)\equiv 1$ as well.