Spectral analysis of one-term symmetric differential operators of even order with interior singularity

TL;DR

This paper investigates the spectral properties of symmetric differential operators of even order with interior singularities, focusing on deficiency indices, self-adjoint extensions, and spectrum characterization.

Contribution

It provides a detailed analysis of deficiency numbers, self-adjoint extensions, and spectral properties for a class of singular differential operators with interior zeros.

Findings

Determined deficiency numbers for the operators.

Described all self-adjoint extensions.

Analyzed the spectral properties of these extensions.

Abstract

In this paper we discuss the spectral properties of one-term symmetric differential operators of even order with interior singularity, namely, we determine the deficiency numbers, describe its self-adjoint extensions and their spectrum. It is assumed that the operators are generated by the differential expression $$l_{2m}[y](x)=(-1)^m(c(x)y^{(m)})^{(m)}(x),$$ where $x \in I:=[-1,1]$, the coefficient $c(x)$ has one zero on the set $I$, and the orders of this zero on the right side and the left side are not necessarily equal.