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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Surgery principles for the spectral analysis of quantum graphs

Authors: Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov and Delio Mugnolo
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 34B45; Secondary 05C50, 35P15, 81Q35
Published electronically: June 21, 2019
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Abstract: We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet, or $ \delta $-type) which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include ``transplantation'' of volume within a graph based on the behaviour of its eigenfunctions, as well as ``unfolding'' of local cycles and pendants. In other cases we establish sharp generalisations, extensions, and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions, and introducing new pendant subgraphs.

To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest non-trivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates -- one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) -- and includes them as special cases.

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Additional Information

Gregory Berkolaiko
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

James B. Kennedy
Affiliation: Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, P-1749-016 Lisboa, Portugal

Pavel Kurasov
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

Delio Mugnolo
Affiliation: Lehrgebiet Analysis, Fakultät Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany

Keywords: Quantum graphs, spectral geometry of quantum graphs, bounds on spectral gaps
Received by editor(s): August 3, 2018
Received by editor(s) in revised form: March 11, 2019
Published electronically: June 21, 2019
Additional Notes: The work of the first author was partially supported by the NSF under grant DMS-1410657.
The second author was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program “Investigador FCT”, reference IF/01461/2015, and project PTDC/MAT-CAL/4334/2014.
The third author was partially supported by the Swedish Research Council (Grant D0497301).
The fourth author was partially supported by the Deutsche Forschungsgemeinschaft (Grant 397230547).
All four authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld, Germany, within the framework of the cooperation group on “Discrete and continuous models in the theory of networks”.
Article copyright: © Copyright 2019 American Mathematical Society