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arXiv · 1705.01831

Spectral Theory of Infinite Quantum Graphs

Pavel Exner·Aleksey Kostenko·Mark Malamud·Hagen Neidhardt
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Abstract

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

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Pavel Exner, Aleksey Kostenko, Mark Malamud, Hagen Neidhardt. 2017-05-04. Spectral Theory of Infinite Quantum Graphs. https://doi.org/10.1007/s00023-018-0728-9

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