Analyticity of resolvents of elliptic operators on quantum graphs with small edges
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Analyticity of resolvents of elliptic operators on quantum graphs with small edges
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Advances in mathematics
ISSN
0001-8708
e-ISSN
1090-2082
Volume of the periodical
397
Issue of the periodical within the volume
March
Country of publishing house
US - UNITED STATES
Number of pages
48
Pages from-to
"Article number: 108125"
UT code for WoS article
000793112500018
EID of the result in the Scopus database
2-s2.0-85119352959