Quantum graphs with the Bethe-Sommerfeld property
https://doi.org/10.17586/2220-8054-2017-8-3-305-309
Abstract
In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned δ-coupling at the vertices.
Keywords
periodic quantum graphs,
gap number,
δ-coupling,
rectangular lattice graph,
scale-invariant coupling,
Bethe-Sommerfeld conjecture,
golden mean
Supporting agencies: The research was supported by the Czech Science Foundation (GACR) within the project 17-01706S.
About the Authors
P. Exner
Czech Technical University; Nuclear Physics Institute CAS
Czech Republic
Czech Republic
Doppler Institute for Mathematical Physics and Applied Mathematics; Department of Theoretical Physics
Bˇrehova 7, 11519 Prague; 25068 Reˇz near Prague
O. Turek
Nuclear Physics Institute CAS; Joint Institute for Nuclear Research; Kochi University of Technology
Czech Republic
Czech Republic
Department of Theoretical Physics; Bogoliubov Laboratory of Theoretical Physics; Laboratory for Unified Quantum Devices
25068 Reˇz near Prague; 141980 Dubna; Kochi 782-8502
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Review
For citations:
Exner P., Turek O. Quantum graphs with the Bethe-Sommerfeld property. Nanosystems: Physics, Chemistry, Mathematics. 2017;8(3):305-309. https://doi.org/10.17586/2220-8054-2017-8-3-305-309
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