Povzner-Wienholtz-type theorems for Sturm-Liouville operators with singular coefficients

TL;DR

This paper extends Povzner-Wienholtz theorems to Sturm-Liouville operators with highly singular coefficients, establishing conditions under which semi-bounded operators are self-adjoint.

Contribution

It introduces new sufficient conditions on the coefficient p ensuring semi-bounded Sturm-Liouville operators with singular coefficients are self-adjoint.

Findings

Operators with singular coefficients can be self-adjoint under certain conditions.

Minimal regularity assumptions on coefficients are sufficient for self-adjointness.

The results apply to coefficients that are Radon measures or discontinuous functions.

Abstract

We introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the coefficients. They are assumed to satisfy conditions \[q=Q'+s;\quad \frac{1}{\sqrt{|p|}}, \frac{Q}{\sqrt{|p|}}, \frac{r}{\sqrt{|p|}} \in L^2_{loc}\left(\mathbb{R}\right), \quad s \in L^1_{loc}\left(\mathbb{R}\right), \quad\frac{1}{p}\neq 0\,\,\text{a.e.,} \] where the derivative of the function $Q$ is understood in the sense of distributions, and all functions $p$, $Q$, $r$, $s$ are real-valued. In particular, the coefficients $q$ and $r'$ may be Radon measures on $\mathbb{R}$, while function $p$ may be discontinuous. The main result of the paper are constructive sufficient conditions on the coefficient $p$ which provide that the operator $L_0$ being semi-bounded implies it being self-adjoint.