On the spectrum of a periodic magnetic Dirac operator

We consider the periodic three-dimensional Dirac operator $\widehat {\mathcal D} +\widehat W=\sum \widehat \alpha _j(-i\frac {\partial }{\partial x_j}-A_j)+\widehat V_0+ \widehat V_1$. The vector potential $A\colon {\mathbb R}^3\to {\mathbb R}^3$ and the functions $\widehat V_s$, $s=0,1$, with values in the space of Hermitian $(4\times 4)$-matrices are periodic with a common period lattice $\Lambda \subset {\mathbb R}^3$. The functions $\widehat V_s$ are supposed to satisfy the commutation relations $\widehat V_s\widehat \alpha _j=(-1)^s\widehat \alpha _j\widehat V_s$, $j=1,2,3$, $s=0,1$. Let $K$ be the fundamental domain of the lattice $\Lambda $. We prove absolute continuity of the spectrum of the operator $\widehat {\mathcal D}+\widehat W$ provided that $A\in H^q_{\mathrm {loc}} ({\mathbb R}^3;{\mathbb R}^3)$, $q>1$, or $\sum \| A_N\| <+\infty $ where $A_N$ are the Fourier coefficients of the magnetic potential $A$, and the function $\widehat V=\widehat V_0+ \widehat V_1$ belongs to the space $L^3_w(K)$ and satisfies the estimate ${\mathrm {mes}}\, \{ x\in K:\| \widehat V(x)\| >t\} \leqslant Ct^{-3}$ for all sufficiently large numbers $t>0$. The constant $C>0$ depends on the $A$ (if $A\equiv 0$ then $C$ is a universal constant), and $\mathrm {mes}$ is the Lebesgue measure. We can also add a function of the same form with several Coulomb singularities $|x-x_m|^{-1}\widehat w_m$ in neighborhoods of points $x_m\in K$, $m=1,\ldots ,m_0$, to the function $\widehat V=\widehat V_0+\widehat V_1$ provided that this function is continuous for $x\notin x_m+\Lambda $, $m=1,\ldots ,m_0$, and $\| \widehat w_m\| \leqslant C_1$ for all $m$. The constant $C_1>0$ also depends on the magnetic potential $A$ (and does not depend on the $m_0$).