Normal Extensions of a Singular Multipoint Differential Operator for First Order

TL;DR

This paper characterizes all normal extensions of a singular multipoint differential operator in a Hilbert space framework and analyzes their spectral properties.

Contribution

It provides a complete description of all normal extensions of a specific multipoint differential operator in Hilbert spaces, including boundary conditions and spectral structure.

Findings

All normal extensions are described via boundary values.

The spectral structure of these extensions is analyzed.

The work advances understanding of multipoint differential operators.

Abstract

In this work, firstly in the direct sum of Hilbert spaces of vector-functions $L^{2} (H,(-\infty,a_{1})) \oplus L^{2} (H,(a_{2},b_{2}))\oplus^{2} (H,(a_{3},+\infty))$, $- \infty<a_{1}<a_{2}<b_{2}<a_{3}<+\infty$ all normal extensions of the minimal operator generated by linear singular multipoint formally normal differential expression $l=(l_{1},l_{2},l_{3}),l_{k} = \frac{d}{dt}+A_{k}$ 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.