Selfadjoint Extensions of a Singular Multipoint Differential Operator for First Order

TL;DR

This paper characterizes all selfadjoint extensions of a singular multipoint differential operator in a Hilbert space and analyzes their spectral properties, advancing the understanding of boundary value problems for such operators.

Contribution

It provides a complete description of all selfadjoint extensions of a multipoint differential operator with operator coefficients using boundary values.

Findings

All selfadjoint extensions are characterized in terms of boundary conditions.

The spectral structure of these extensions is thoroughly investigated.

Results apply to operators with operator coefficients in Hilbert spaces.

Abstract

In this work, firstly in the direct sum of Hilbert spaces of vector-functions L^2 (H,(-{\infty},a_1)){\Box}L^2 (H,(a_2,b_2)){\Box}L^2 (H,(a_3,+{\infty})),- {\infty}<a_1<a_2<b_2<a_3<+{\infty} all selfadjoint extensions of the minimal operator generated by linear singular multipoint symmetric differential expression l=(l_1,l_2,l_3),l_k=i d/dt+A_k with a selfadjoint operator coefficient A_k k=1,2,3 in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.