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

L. I. Danilov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 3-26

Résumé

This paper is concerned with a two-dimensional Dirac operator $\widehat \sigma _1\bigl( -i\, \frac {\partial }{\partial x_1}\bigr) +\widehat \sigma _2\bigl( -i\, \frac {\partial }{\partial x_2}-Bx_1\bigr) +m\widehat \sigma _3+V\widehat I_2$ 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.

Détail

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