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 Izv. IMI UdGU, 2019, Volume 54, Pages 3–26 (Mi iimi378)

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

On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential

L. I. Danilov

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

Abstract: This paper is concerned with a two-dimensional Dirac operator $\widehat \sigma _1( -i \frac {\partial }{\partial x_1}) +\widehat \sigma _2( -i \frac {\partial }{\partial x_2}-Bx_1) +m\widehat \sigma _3+V\widehat I_2$ with a uniform magnetic field $B$ where $\widehat \sigma _j$, $j=1,2,3$, are the Pauli matrices and $\widehat I_2$ is the unit $2\times 2$-matrix. The function $m$ and the electric potential $V$ belong to the space $L^p_{\Lambda }({\mathbb R}^2;{\mathbb R})$ of $\Lambda$-periodic functions from the $L^p_{\mathrm {loc}}({\mathbb R}^2;{\mathbb R})$, $p>2$, and we suppose that for the magnetic flux $\eta =(2\pi )^{-1}Bv(K)\in \mathbb{Q}$ where $v(K)$ is the area of an elementary cell $K$ of the period lattice $\Lambda$. For any nonincreasing function $(0,1]\ni \varepsilon \mapsto {\mathcal R}(\varepsilon )\in (0,+\infty )$ for which ${\mathcal R}(\varepsilon )\to +\infty$ as $\varepsilon \to +0$ let ${\mathfrak M}^p_{\Lambda }({\mathcal R}(\cdot ))$ be the set of functions $m\in L^p_{\Lambda }({\mathbb R}^2;{\mathbb R})$ such that for every $\varepsilon \in (0,1]$ there exists a real-valued $\Lambda$-periodic trigonometric polynomial ${\mathcal P}^{(\varepsilon )}$ such that $\| m-{\mathcal P} ^{(\varepsilon )}\| _{L^p(K)}<\varepsilon$ and for Fourier coefficients ${\mathcal P}^{(\varepsilon )}_Y=0$ provided $|Y|>{\mathcal R}(\varepsilon )$. It is proved that for any function ${\mathcal R}(\cdot )$ in question there is a dense $G_{\delta }$-set ${\mathcal O}$ in the Banach space $(L^p_{\Lambda }({\mathbb R}^2;{\mathbb R}),\| \cdot \| _{L^p(K)})$ such that for every electric potential $V\in {\mathcal O}$, for every function $m\in {\mathfrak M}^p_{\Lambda }({\mathcal R} (\cdot ))$, and for every uniform magnetic field $B$ with the flux $\eta \in \mathbb{Q}$ the spectrum of the Dirac operator is absolutely continuous.

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

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations AAAA-A16-116021010082-8 The study was funded by the financing program AAAA-A16-116021010082-8.

DOI: https://doi.org/10.20537/2226-3594-2019-54-01

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Bibliographic databases:

UDC: 517.958, 517.984.56
MSC: 35P05
Received: 24.10.2019

Citation: L. I. Danilov, “On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential”, Izv. IMI UdGU, 54 (2019), 3–26

Citation in format AMSBIB
\Bibitem{Dan19} \by L.~I.~Danilov \paper On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential \jour Izv. IMI UdGU \yr 2019 \vol 54 \pages 3--26 \mathnet{http://mi.mathnet.ru/iimi378} \crossref{https://doi.org/10.20537/2226-3594-2019-54-01} \elib{https://elibrary.ru/item.asp?id=41435137} 

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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
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