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 Izv. IMI UdGU, 2020, Volume 55, Pages 42–59 (Mi iimi390)

MATHEMATICS

On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$

L. I. Danilov

Udmurt Federal Research Center, Ural Branch of the Russian Academy of Sciences, ul. T. Baramzinoi, 34, Izhevsk, 426067, Russia

Abstract: We consider the two-dimensional Shrödinger operator $\widehat H_B+V$ with a homogeneous magnetic field $B\in {\mathbb R}$ and with an electric potential $V$ which belongs to the space $L^p_{\Lambda } ({\mathbb R}^2;{\mathbb R})$ of $\Lambda$ -periodic real-valued functions from the space $L^p_{\mathrm {loc}} ({\mathbb R}^2)$, $p>1$. The magnetic field $B$ is supposed to have the rational flux $\eta =(2\pi )^{-1}Bv(K) \in {\mathbb Q}$ where $v(K)$ denotes the area of the elementary cell $K$ of the period lattice $\Lambda \subset {\mathbb R}^2$. Given $p>1$ and the period lattice $\Lambda$, we prove that in the Banach space $(L^p_{\Lambda } ({\mathbb R}^2;\mathbb R),\| \cdot \| _{L^p(K)})$ there exists a typical set $\mathcal O$ in the sense of Baire (which contains a dense $G_{\delta}$ -set) such that the spectrum of the operator $\widehat H_B+V$ is absolutely continuous for any electric potential $V\in {\mathcal O}$ and for any homogeneous magnetic field $B$ with the rational flux $\eta \in {\mathbb Q}$.

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

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations AAAA-A17-117022250041-7 The research is supported by the financial program AAAA-A17-117022250041-7.

DOI: https://doi.org/10.35634/2226-3594-2020-55-04

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Bibliographic databases:

UDC: 517.958, 517.984.56
MSC: 35P05

Citation: L. I. Danilov, “On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$”, Izv. IMI UdGU, 55 (2020), 42–59

Citation in format AMSBIB
\Bibitem{Dan20} \by L.~I.~Danilov \paper On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$ \jour Izv. IMI UdGU \yr 2020 \vol 55 \pages 42--59 \mathnet{http://mi.mathnet.ru/iimi390} \crossref{https://doi.org/10.35634/2226-3594-2020-55-04} \elib{https://elibrary.ru/item.asp?id=42949300}