On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter,...

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Bibliographic Details
Published in:Annales Henri Poincaré 2016, Vol.17 (9), p.2439-2473
Main Authors: Kennedy, James B., Kurasov, Pavel, Malenová, Gabriela, Mugnolo, Delio
Format: Article
Language:English
Subjects:
Apexes
Boundary conditions
Classical and Quantum Gravitation
Combinatorial analysis
Dynamical Systems and Ergodic Theory
Elementary Particles
Graph theory
Graphs
Lower bounds
Mathematical and Computational Physics
Mathematical Methods in Physics
Parameter estimation
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
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Summary:We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.
ISSN:1424-0637
1424-0661
1424-0661
DOI:10.1007/s00023-016-0460-2