Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

TL;DR

This paper introduces a method to reduce the spectral analysis of certain self-adjoint extensions, especially on graph-like structures, to simpler problems involving bounded operators, facilitating analysis of differential operators on graphs.

Contribution

It provides a novel unitary reduction technique for self-adjoint extensions with specific Weyl functions, applicable to differential operators on metric graphs.

Findings

Spectral problems can be reduced to bounded operator problems in certain cases.

A class of boundary conditions enables unitary reduction to generalized discrete Laplacians.

Application to differential operators on equilateral metric graphs demonstrates practical utility.

Abstract

We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form $M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians.