On some class of singular Sturm-Liouville problems

TL;DR

This paper investigates a class of singular Sturm-Liouville problems involving generalized functions, demonstrating their reduction to simpler forms and exploring specific cases with self-similar coefficients.

Contribution

It introduces a method to reduce complex singular Sturm-Liouville problems to standard forms and analyzes special cases with self-similar coefficients.

Findings

Reduction of singular problems to standard form with r ≡ 1

Analysis of q=0 case with self-similar r and p

Illustration of the reduction method with examples

Abstract

Sturm-Liouville spectral problem for equation $-(y'/r)'+qy=\lambda py$ with generalized functions $r\ge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $r\equiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.