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Trace formulae for quantum graphs. (English) Zbl 1153.81493

Exner, Pavel (ed.) et al., Analysis on graphs and its applications. Selected papers based on the Isaac Newton Institute for Mathematical Sciences programme, Cambridge, UK, January 8–June 29, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4471-7/hbk). Proceedings of Symposia in Pure Mathematics 77, 247-259 (2008).
Summary: Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of the spectral density in terms of sums over periodic orbits. Finally, a general trace formula for any self adjoint realisation of the Laplacian on a compact, metric graph is given.
For the entire collection see [Zbl 1143.05002].

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q50 Quantum chaos
34B45 Boundary value problems on graphs and networks for ordinary differential equations
46A99 Topological linear spaces and related structures
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