Spectra of Elliptic Operators on Quantum Graphs with Small Edges

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F21%3A50018608" target="_blank" >RIV/62690094:18470/21:50018608 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.mdpi.com/2227-7390/9/16/1874" target="_blank" >https://www.mdpi.com/2227-7390/9/16/1874</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.3390/math9161874" target="_blank" >10.3390/math9161874</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Spectra of Elliptic Operators on Quantum Graphs with Small Edges

Popis výsledku v původním jazyce

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 gamma by a small positive parameter epsilon. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on epsilon and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph gamma 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 epsilon 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.

Název v anglickém jazyce

Spectra of Elliptic Operators on Quantum Graphs with Small Edges

Popis výsledku anglicky

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 gamma by a small positive parameter epsilon. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on epsilon and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph gamma 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 epsilon 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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

MATHEMATICS

ISSN

2227-7390

e-ISSN

—

Svazek periodika

9

Číslo periodika v rámci svazku

16

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

24

Strana od-do

"Article Number: 1874"

Kód UT WoS článku

000689404600001

EID výsledku v databázi Scopus

2-s2.0-85112396760