Peierls' substitution for low lying spectral energy windows

Horia Cornean, Bernard Helffer, Radu Purice

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2 Citations (Scopus)


We consider a 2d periodic magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a “Landau type” structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

Original languageEnglish
JournalJournal of Spectral Theory
Issue number4
Pages (from-to)1179-1222
Number of pages44
Publication statusPublished - 2019


  • Magnetic pseudo-differential operators, spectral gaps, Wannier functions
  • Magnetic pseudo-differential operators
  • Spectral gaps
  • Wannier functions

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