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Annales Henri Poincaré

, Volume 19, Issue 11, pp 3457–3510 | Cite as

Spectral Theory of Infinite Quantum Graphs

  • Pavel Exner
  • Aleksey Kostenko
  • Mark Malamud
  • Hagen Neidhardt
Open Access
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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.

Notes

Acknowledgements

Open access funding provided by University of Vienna. We thank Noema Nicolussi for useful discussions and helpful remarks. We are also grateful to the referees for the careful reading of our manuscript, their remarks and hints with respect to the literature that have helped to improve the exposition. A.K. appreciates the hospitality at the Department of Theoretical Physics, Nuclear Physics Institute, during several short stays in 2016, where a part of this work was done.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Doppler Institute for Mathematical Physics and Applied MathematicsCzech Technical UniversityPragueCzech Republic
  2. 2.Department of Theoretical Physics Nuclear Physics InstituteCzech Academy of SciencesŘež, PragueCzech Republic
  3. 3.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  4. 4.Faculty of MathematicsUniversity of ViennaViennaAustria
  5. 5.Peoples Friendship University of Russia (RUDN University)MoscowRussian Federation
  6. 6.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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