Isospectrality for graph laplacians under the change of coupling at graph vertices

Yulia Ershova; Irina I. Karpenko; Alexander V. Kiselev

doi:10.4171/JST/117

Isospectrality for graph laplacians under the change of coupling at graph vertices

Yulia Ershova, Irina I. Karpenko, Alexander V. Kiselev

Research output: Contribution to journal › Article › peer-review

12 Citations (SciVal)

Abstract

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of δ and δ' types. An infinite set of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrödinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.

Original language	English
Pages (from-to)	43-66
Number of pages	24
Journal	Journal of Spectral Theory
Volume	6
Issue number	1
DOIs	https://doi.org/10.4171/JST/117
Publication status	Published - 2016

Bibliographical note

Funding Information:
The third author's work was partially supported by the RFBR, grant no. 12-01-00215-a and by the grant of the Ministry of Science of Ukraine.

Publisher Copyright:
© 2016 European Mathematical Society.

Keywords

Boundary triples
Inverse spectral problem
Isospectral graphs
Laplace operator
Quantum graphs
Schrödinger operator
Trace formulae

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Geometry and Topology

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10.4171/JST/117

Cite this

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title = "Isospectrality for graph laplacians under the change of coupling at graph vertices",

abstract = "Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of δ and δ' types. An infinite set of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schr{\"o}dinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.",

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author = "Yulia Ershova and Karpenko, {Irina I.} and Kiselev, {Alexander V.}",

note = "Funding Information: The third author's work was partially supported by the RFBR, grant no. 12-01-00215-a and by the grant of the Ministry of Science of Ukraine. Publisher Copyright: {\textcopyright} 2016 European Mathematical Society.",

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doi = "10.4171/JST/117",

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AU - Ershova, Yulia

AU - Karpenko, Irina I.

AU - Kiselev, Alexander V.

N1 - Funding Information: The third author's work was partially supported by the RFBR, grant no. 12-01-00215-a and by the grant of the Ministry of Science of Ukraine. Publisher Copyright: © 2016 European Mathematical Society.

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N2 - Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of δ and δ' types. An infinite set of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrödinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.

AB - Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of δ and δ' types. An infinite set of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrödinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.

KW - Boundary triples

KW - Inverse spectral problem

KW - Isospectral graphs

KW - Laplace operator

KW - Quantum graphs

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Isospectrality for graph laplacians under the change of coupling at graph vertices

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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