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Spectral Theory of Infinite Quantum Graphs

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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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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Correspondence to Aleksey Kostenko.

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Communicated by Jan Derezinski.

Dedicated to the memory of M. Z. Solomyak (16.05.1931–31.07.2016).

Research supported by the Czech Science Foundation (GAČR) under Grant No. 17-01706S and the European Union within the Project CZ.02.1.01/0.0/0.0/16_019/0000778 (P.E.), by the Austrian Science Fund (FWF) under Grant No. P28807 (A.K.), by the “RUDN University Program 5-100” (M.M.) and by the European Research Council (ERC) under Grant No. AdG 267802 “AnaMultiScale” (H.N.).

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Exner, P., Kostenko, A., Malamud, M. et al. Spectral Theory of Infinite Quantum Graphs. Ann. Henri Poincaré 19, 3457–3510 (2018). https://doi.org/10.1007/s00023-018-0728-9

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