Periodic quantum graphs with predefined spectral gaps

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Let Γ be an arbitrary periodic metric graph, which does not coincide with a line. We consider the Hamiltonian on its edges; here ϵ > 0 is a small parameter. Let. We show that under a proper choice of vertex conditions the spectrum has at least m gaps as ϵ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of as ϵ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ϵ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.

Original languageEnglish
Article number405202
JournalJournal of Physics A: Mathematical and Theoretical
Issue number40
Publication statusPublished - 9 Oct 2020


  • Control of spectrum
  • Periodic quantum graphs
  • Spectral gaps
  • Î-interactions
  • Î_-interactions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)


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