Spectra of Elliptic Operators on Quantum Graphs with Small Edges

Denis I. Borisov

doi:10.3390/math9161874

Abstract

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions.

Highlights

The limiting operators was defined on a graph, in which the small edges were replaced by the vertices to which they shrink and at such vertices certain limiting boundary conditions were determined
We prove the analyticity of the perturbed eigenvalues and the associated eigenfunctions, while in the fifth section, we describe an algorithm for determining the coefficients of their Taylor series
The first of them concerns the analyticity of the eigenvalues

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The limiting operators was defined on a graph, in which the small edges were replaced by the vertices to which they shrink and at such vertices certain limiting boundary conditions were determined. These results were further developed in [6,7], where a general elliptic operator with varying coefficients was considered on a graph with small edges rescaled by means of a single small parameter. We prove the analyticity of the perturbed eigenvalues and the associated eigenfunctions, while, we describe an algorithm for determining the coefficients of their Taylor series We prove the analyticity of the perturbed eigenvalues and the associated eigenfunctions, while in the fifth section, we describe an algorithm for determining the coefficients of their Taylor series

Problem

Assumptions and Notations

Main Results

Discussion and Possible Generalizations of Main Results

Convergence of Spectrum

Taylor Series

Conclusions