Normal Extensions of an Singular Differential Operator for First Order

TL;DR

This paper characterizes all normal extensions of a minimal singular differential operator with selfadjoint coefficients in a Hilbert space, describing their boundary conditions and analyzing their spectral properties.

Contribution

It provides a comprehensive description of all normal extensions of a specific singular differential operator with selfadjoint coefficients in a Hilbert space.

Findings

All normal extensions are described via boundary values.

The spectral structure of these extensions is analyzed.

Results apply to operators with selfadjoint coefficients in Hilbert spaces.

Abstract

In this work, in the Hilbert space of vector-functions L^2 (H,(-\infty,a)\cup(b,+\infty)),a<b all normal extensions of the minimal operator generated by linear singular formally normal differential expression l(\cdot)=(d/dt+A_1,d/dt+A_2) with a selfadjoint operator coefficients A_1 andA_2 in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.