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This article is cited in 3 scientific papers (total in 3 papers)
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.
Received: 01.05.2020
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
Linking options:
https://www.mathnet.ru/eng/iimi390 https://www.mathnet.ru/eng/iimi/v55/p42
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