Spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat D = \sum\limits_{j - 1}^n {\left( { - i\frac{\partial }{{\partial x_j }} - A_j } \right)} \hat \alpha _j + \hat V^{\left( 0 \right)} + \hat V^{\left( 1 \right)} ,x \in R^n ,n \geqslant 3,$$ \end{document}, is proved given that A∈C(Rn;Rn)⊂Hlocq(Rn;Rn), 2q>n−2, and also that the Fourier series of the vector potential A:Rn→Rn is absolutely convergent. Here,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat V^{\left( s \right)} = \left( {\hat V^{\left( s \right)} } \right)^* $$ \end{document} are continuous matrix functions and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat V^{\left( s \right)} \hat \alpha _j = \left( { - 1} \right)^{\left( s \right)} \hat \alpha _j \hat V^{\left( s \right)} $$ \end{document} for all anticommuting Hermitian matrices\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat \alpha _j ,\hat \alpha _j^2 = \hat I,s = 0,1$$ \end{document}.