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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

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

Name of the periodical

UFA MATHEMATICAL JOURNAL

ISSN

2074-1863

e-ISSN

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Volume of the periodical

11

Issue of the periodical within the volume

2

Country of publishing house

RU - RUSSIAN FEDERATION

Number of pages

15

Pages from-to

56-70

UT code for WoS article

000511171600004

EID of the result in the Scopus database

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