Selfadjoint extensions of a singular differential operator

TL;DR

This paper characterizes all selfadjoint extensions of a singular differential operator with a selfadjoint coefficient in a Hilbert space, and analyzes their spectral properties, advancing understanding of boundary value problems in quantum mechanics.

Contribution

It provides a complete description of all selfadjoint extensions of a specific singular differential operator in terms of boundary values, including spectral structure analysis.

Findings

All selfadjoint extensions are described via boundary conditions.

Spectral properties of these extensions are thoroughly investigated.

The work advances boundary value problem theory for singular differential operators.

Abstract

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