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Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case

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The goal is to investigate spectral properties of the operator H=(−i∇ +a(x))2+a0(x) in the two-dimensional situation, a(x), a0(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.

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Authors and Affiliations

  1. Department of Mathematics, CH452, University of Alabama at Birmingham, Birmingham, AL, 35294-1170, USA

    Yulia Karpeshina

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  1. Yulia Karpeshina
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Correspondence to Yulia Karpeshina.

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Communicated by B. Simon

Research partially supported by USNSF Grant DMS-0201383.

Acknowledgements The author is thankful to Konstantin Makarov for very useful discussions and to Young-Ran Lee for her great help with pictures.

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Karpeshina, Y. Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case. Commun. Math. Phys. 251, 473–514 (2004). https://doi.org/10.1007/s00220-004-1129-0

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