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

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.