SIMPLEST GRAPHS WITH SMALL EDGES: ASYMPTOTICS FOR RESOLVENTS AND HOLOMORPHIC DEPENDENCE OF SPECTRUM

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F19%3A50017090" target="_blank" >RIV/62690094:18470/19:50017090 - isvavai.cz</a>

Výsledek na webu

<a href="http://matem.anrb.ru/sites/default/files/files/vupe42/Borisov.pdf" target="_blank" >http://matem.anrb.ru/sites/default/files/files/vupe42/Borisov.pdf</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.13108/2019-11-2-56" target="_blank" >10.13108/2019-11-2-56</a>

Jazyk výsledku

angličtina

Název v původním jazyce

SIMPLEST GRAPHS WITH SMALL EDGES: ASYMPTOTICS FOR RESOLVENTS AND HOLOMORPHIC DEPENDENCE OF SPECTRUM

Popis výsledku v původním jazyce

In the work we consider a simplest graph formed by two finite edges and a small edge coupled at a common vertex. The length of the small edge serves as a small parameter. On such graph, we consider the Schrodinger operator with the Kirchoff condition at the internal vertex, the Dirichlet condition on the boundary vertices of finite edges and the Dirichlet or Neumann condition on the boundary vertex of the small edge. We show that such operator converges to a Schrodinger operator on the graph without the small edge in the norm resolvent sense; at the internal vertex one has to impose the Dirichlet condition if the same was on the boundary vertex of the small edge. If the boundary vertex was subject to the Neumann condition, the internal vertex keeps the Kirchoff condition but the coupling constant can change. The main obtained result for the resolvents is the two-terms asymptotics for their resolvents and an estimate for the error term. The second part of the work is devoted to studying the dependence of the eigenvalues on the small parameter. Despite the graph is perturbed singularly, the eigenvalues are holomorphic in the small parameter and are represented by convergent series. We also find out that under the perturbation, there can be stable eigenvalues independent of the parameter. We provide a criterion determining the existence of such eigenvalues. For varying eigenvalues we find the leading terms of their Taylor series.

Název v anglickém jazyce

SIMPLEST GRAPHS WITH SMALL EDGES: ASYMPTOTICS FOR RESOLVENTS AND HOLOMORPHIC DEPENDENCE OF SPECTRUM

Popis výsledku anglicky

In the work we consider a simplest graph formed by two finite edges and a small edge coupled at a common vertex. The length of the small edge serves as a small parameter. On such graph, we consider the Schrodinger operator with the Kirchoff condition at the internal vertex, the Dirichlet condition on the boundary vertices of finite edges and the Dirichlet or Neumann condition on the boundary vertex of the small edge. We show that such operator converges to a Schrodinger operator on the graph without the small edge in the norm resolvent sense; at the internal vertex one has to impose the Dirichlet condition if the same was on the boundary vertex of the small edge. If the boundary vertex was subject to the Neumann condition, the internal vertex keeps the Kirchoff condition but the coupling constant can change. The main obtained result for the resolvents is the two-terms asymptotics for their resolvents and an estimate for the error term. The second part of the work is devoted to studying the dependence of the eigenvalues on the small parameter. Despite the graph is perturbed singularly, the eigenvalues are holomorphic in the small parameter and are represented by convergent series. We also find out that under the perturbation, there can be stable eigenvalues independent of the parameter. We provide a criterion determining the existence of such eigenvalues. For varying eigenvalues we find the leading terms of their Taylor series.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název periodika

UFA MATHEMATICAL JOURNAL

ISSN

2074-1863

e-ISSN

—

Svazek periodika

11

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

RU - Ruská federace

Počet stran výsledku

15

Strana od-do

56-70

Kód UT WoS článku

000511171600004

EID výsledku v databázi Scopus

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