Regularization of singular Sturm-Liouville equations

TL;DR

This paper introduces a new regularization method for singular Sturm-Liouville equations with distributional coefficients, enabling proper operator definition and analysis of their extensions and resolvent approximations.

Contribution

It develops a novel regularization approach that correctly defines the operators as quasi-differential and describes their extensions and resolvents in canonical boundary forms.

Findings

Operators are correctly defined as quasi-differential.

Self-adjoint and dissipative extensions are characterized by boundary conditions.

Results include new insights even for the case when p(t) ≡ 1.

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.