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Periodic quantum graphs from the Bethe-Sommerfeld perspective

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F17%3A00486254" target="_blank" >RIV/61389005:_____/17:00486254 - isvavai.cz</a>

  • Alternative codes found

    RIV/68407700:21340/17:00319053

  • Result on the web

    <a href="http://dx.doi.org/10.1088/1751-8121/aa8d8d" target="_blank" >http://dx.doi.org/10.1088/1751-8121/aa8d8d</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1088/1751-8121/aa8d8d" target="_blank" >10.1088/1751-8121/aa8d8d</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Periodic quantum graphs from the Bethe-Sommerfeld perspective

  • Original language description

    The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To date, its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the 'Bethe-Sommerfeld property', that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a delta coupling at the vertices and a suitable irrational ratio of the edges. Our result allows one to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to date.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • OECD FORD branch

    10301 - Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)

Result continuities

  • Project

    <a href="/en/project/GA17-01706S" target="_blank" >GA17-01706S: Mathematical-Physics Models of Novel Materials</a><br>

  • Continuities

    P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Others

  • Publication year

    2017

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Journal of Physics A-Mathematical and Theoretical

  • ISSN

    1751-8113

  • e-ISSN

  • Volume of the periodical

    50

  • Issue of the periodical within the volume

    45

  • Country of publishing house

    GB - UNITED KINGDOM

  • Number of pages

    32

  • Pages from-to

  • UT code for WoS article

    000423284300001

  • EID of the result in the Scopus database

    2-s2.0-85032215939