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Spectrum of the Landau Hamiltonian with a Periodic Electric Potential

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Abstract

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrödinger operator is absolutely continuous in the case of a homogeneous magnetic field B with a rational flux η = (2π)−1Bυ(K), where υ(K) is the area of an elementary cell K in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in L2loc(ℝ2) with a given period lattice and identified with L2(K), there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow η, the spectrum of the two-dimensional Schrödinger operator is absolutely continuous.

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Authors and Affiliations

  1. Udmurt Federal Research Center, Ural Branch, RAS, Izhevsk, Russia

    L. I. Danilov

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  1. L. I. Danilov
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Correspondence to L. I. Danilov.

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Conflicts of interest. The author declares no conflicts of interest.

Additional information

This research is supported by the financial program AAAA-A16-116021010082-8.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 1, pp. 47–65, January, 2020.

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Danilov, L.I. Spectrum of the Landau Hamiltonian with a Periodic Electric Potential. Theor Math Phys 202, 41–57 (2020). https://doi.org/10.1134/S0040577920010055

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  • DOI: https://doi.org/10.1134/S0040577920010055

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