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TMF, 2020, Volume 202, Number 1, Pages 47–65 (Mi tmf9748)  

This article is cited in 1 scientific paper (total in 1 paper)

Spectrum of the Landau Hamiltonian with a periodic electric potential

L. I. Danilov

Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, Izhevsk, Russia

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 $\eta= (2\pi)^{-1}Bv(K)$, where $v(K)$ 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 $L^2_{\mathrm{loc}}(\mathbb R^2)$ with a given period lattice and identified with $L^2(K)$, 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 $\eta$, the spectrum of the two-dimensional Schrödinger operator is absolutely continuous.

Keywords: two-dimensional Schrödinger operator, absolute spectrum continuity, periodic potential, homogeneous magnetic field.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations AAAA-A16-116021010082-8
This research is supported by the financial program AAAA-A16-116021010082-8.


DOI: https://doi.org/10.4213/tmf9748

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English version:
Theoretical and Mathematical Physics, 2020, 202:1, 41–57

Bibliographic databases:

MSC: 35P05
Received: 15.05.2019
Revised: 15.05.2019

Citation: L. I. Danilov, “Spectrum of the Landau Hamiltonian with a periodic electric potential”, TMF, 202:1 (2020), 47–65; Theoret. and Math. Phys., 202:1 (2020), 41–57

Citation in format AMSBIB
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\paper Spectrum of the~Landau Hamiltonian with a~periodic electric potential
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\pages 41--57
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. I. Danilov, “O spektre gamiltoniana Landau s periodicheskim elektricheskim potentsialom $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$”, Izv. IMI UdGU, 55 (2020), 42–59  mathnet  crossref  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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