Spectral theory of infinite quantum graphs

Pavel Exner Aleksey Kostenko Mark Malamud Hagen Neidhardt

Combinatorics mathscidoc:1910.43179

Annales Henri Poincar, 19, (11), 3457-3510, 2018.11
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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  title={Spectral theory of infinite quantum graphs},
  author={Pavel Exner, Aleksey Kostenko, Mark Malamud, and Hagen Neidhardt},
  booktitle={Annales Henri Poincar},
Pavel Exner, Aleksey Kostenko, Mark Malamud, and Hagen Neidhardt. Spectral theory of infinite quantum graphs. 2018. Vol. 19. In Annales Henri Poincar. pp.3457-3510. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020132449980260708.
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