Spectrum of the periodic Dirac operator

Abstract

The absolute continuity of the spectrum for the periodic Dirac operator

$$\hat D = \sum\limits_{j - 1}^n {\left( { - i\frac{\partial }{{\partial x_j }} - A_j } \right)} \hat \alpha _j + \hat V^{\left( 0 \right)} + \hat V^{\left( 1 \right)} ,x \in R^n ,n \geqslant 3,$$

, is proved given that A∈C(R ⁿ;R ⁿ)⊂H _loc ^q(R ⁿ;R ⁿ), 2q>n−2, and also that the Fourier series of the vector potential A:R ⁿ→R ⁿ is absolutely convergent. Here,$\hat V^{\left( s \right)} = \left( {\hat V^{\left( s \right)} } \right)^*$ are continuous matrix functions and$\hat V^{\left( s \right)} \hat \alpha _j = \left( { - 1} \right)^{\left( s \right)} \hat \alpha _j \hat V^{\left( s \right)}$ for all anticommuting Hermitian matrices$\hat \alpha _j ,\hat \alpha _j^2 = \hat I,s = 0,1$.