Annales Henri Poincaré

, Volume 15, Issue 12, pp 2321–2377 | Cite as

Eigenvalues of a One-Dimensional Dirac Operator Pencil

  • Daniel M. Elton
  • Michael Levitin
  • Iosif Polterovich


We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a non-trivial square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.


Potential Versus Dirac Operator Complex Eigenvalue Phase Plot Discrete Subset 
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© Springer Basel 2013

Authors and Affiliations

  • Daniel M. Elton
    • 1
  • Michael Levitin
    • 2
  • Iosif Polterovich
    • 3
  1. 1.Department of Mathematics and StatisticsFylde College, Lancaster UniversityLancasterUK
  2. 2.Department of Mathematics and StatisticsUniversity of Reading, WhiteknightsReadingUK
  3. 3.Département de mathématiques et de statistiqueUniversité de MontréalMontrealCanada

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