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SURGERY PRINCIPLES FOR THE SPECTRAL ANALYSIS OF QUANTUM GRAPHS
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 42019 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 372, no 7, p. 5153-5197Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2019. Vol. 372, no 7, p. 5153-5197
Keywords [en]
Quantum graphs, spectral geometry of quantum graphs, bounds on spectral gaps
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-175026DOI: 10.1090/tran/7864ISI: 000487086300020OAI: oai:DiVA.org:su-175026DiVA, id: diva2:1367131
Available from: 2019-11-01 Created: 2019-11-01 Last updated: 2019-11-01Bibliographically approved

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