This article is cited in
2 papers
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 $\widehat \sigma _1\bigl( -i\,
\frac {\partial }{\partial x_1}\bigr) +\widehat \sigma _2\bigl( -i\, \frac {\partial }{\partial x_2}-Bx_1\bigr)
+m\widehat \sigma _3+V\widehat I_2$ 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.
Keywords:
two-dimensional Dirac operator, periodic electric potential, homogeneous magnetic field,
spectrum.
UDC:
517.958,
517.984.56
MSC: 35P05 Received: 24.10.2019
DOI:
10.20537/2226-3594-2019-54-01
Fulltext:
PDF file (302 kB)