On absolute continuity of the spectrum of three-dimensional periodic Dirac operator

L. I. Danilov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 35 (2006) no. 1, pp. 49-76

Résumé

We prove the absolute continuity of the spectrum of periodic Dirac operator $\sum\limits_{j=1}^3\hat \alpha _j\bigl( -i\, \frac {\partial}{\partial x_j}-A_j\bigr) +\hat {\mathcal V}^{(0)}+\hat {\mathcal V}^{(1)}\, ,\ x\in {\mathbb{R}}^3$, with period lattice $\Lambda \subset {\mathbb{R}}^3$ if $A\in L^{\infty}({\mathbb{R}}^3; {\mathbb{R}}^3)$, $\| \, |A|\, \| _{L^{\infty}({\mathbb{R}}^3)}\max\limits_{\gamma \in \Lambda \backslash \{ 0\} }\pi |\gamma |^{-1}$, the Hermitian matrix-valued functions $\hat {\mathcal V}^{(s)}_{}$ belong to Zigmund class $L^3\ln ^{2+\delta}_{}L(K)$ for some $\delta >0$, where $K$ is the unit cell of the lattice $\Lambda$, and $\hat {\mathcal V}^{(s)}\hat \alpha _j=(-1)^s\hat \alpha _j\hat {\mathcal V}^{(s)}$, $s=0,1$, for all anticommuting Hermitian matrices $\hat \alpha _j^{}\, $, $\hat \alpha _j^2=\hat I$, j=1, 2, 3.