Analyticity of resolvents 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%2F22%3A50018616" target="_blank" >RIV/62690094:18470/22:50018616 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.sciencedirect.com/science/article/pii/S0001870821005648?via%3Dihub" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0001870821005648?via%3Dihub</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.aim.2021.108125" target="_blank" >10.1016/j.aim.2021.108125</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Analyticity of resolvents of elliptic operators on quantum graphs with small edges

Popis výsledku v původním jazyce

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to ε, where ε is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator Hε with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in ε. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of Hε are analytic in ε. This allows us to represent the resolvent of Hε by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of ε. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in ε-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic ε-dependence of the coefficients in the differential expression and vertex conditions. © 2021 Elsevier Inc.

Název v anglickém jazyce

Analyticity of resolvents of elliptic operators on quantum graphs with small edges

Popis výsledku anglicky

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to ε, where ε is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator Hε with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in ε. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of Hε are analytic in ε. This allows us to represent the resolvent of Hε by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of ε. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in ε-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic ε-dependence of the coefficients in the differential expression and vertex conditions. © 2021 Elsevier Inc.

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

2022

Kód důvěrnosti údajů

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

Název periodika

Advances in mathematics

ISSN

0001-8708

e-ISSN

1090-2082

Svazek periodika

397

Číslo periodika v rámci svazku

March

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

48

Strana od-do

"Article number: 108125"

Kód UT WoS článku

000793112500018

EID výsledku v databázi Scopus

2-s2.0-85119352959