najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2017-8-3-305-309najo-599Research ArticleMATHEMATICSМАТЕМАТИКАQuantum graphs with the Bethe-Sommerfeld propertyExnerP.

Doppler Institute for Mathematical Physics and Applied Mathematics; Department of Theoretical Physics

Bˇrehova 7, 11519 Prague; 25068 Reˇz near Prague

exner@ujf.cas.czTurekO.

Department of Theoretical Physics; Bogoliubov Laboratory of Theoretical Physics; Laboratory for Unified Quantum Devices

25068 Reˇz near Prague; 141980 Dubna; Kochi 782-8502

o.turek@ujf.cas.czCzech Technical University; Nuclear Physics Institute CASCzech RepublicNuclear Physics Institute CAS; Joint Institute for Nuclear Research; Kochi University of TechnologyCzech Republic20171108202583305309Copyright © Exner P., Turek O., 20252025Exner P., Turek O.Exner P., Turek O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/599

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.

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The authors declare that there are no conflicts of interest present.