Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups

Details Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups

2006-01-20

arxiv.org/abs/math-ph/0601041v1

Contemporary Mathematics Vol. 415, 2006. p. 201 - 225

Physics

Mathematical Physics

To be published in the Proceedings of the Conference on Quantum Graphs and Their Applications held in Snowbird, Utah, June 18

Scientific paper

The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative eigenvalues and a lower bound on the spectrum of Laplace operators. Also we provide a sufficient condition for the associated heat semigroup to be positivity preserving.

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