Spectrum of the periodic Dirac operator

L. I. Danilov

Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 1, pp. 3-17

Résumé

The absolute continuity of the spectrum for the periodic Dirac operator $$ \widehat D=\sum_{j=1}^n\biggl(-i\frac{\partial}{{\partial}x_j}-A_j\biggr) \widehat\alpha_j+\widehat V^{(0)}+\widehat V^{(1)},\quad x\in\mathbb R^n,\quad n\geq3, $$ is proved given that $A\in C(\mathbb R^n;\mathbb R^n)\cap H_\mathrm{loc}^q(\mathbb R^n;\mathbb R^n)$, $2q>n-2$, and also that the Fourier series of the vector potential $A\colon\mathbb R^n\to\mathbb R^n$ is absolutely convergent. Here, $\widehat V^{(s)}=(\widehat V^{(s)})^*$ are continuous matrix functions and $\widehat V^{(s)}\widehat\alpha_j=(-1)^s\widehat\alpha_j\widehat V^{(s)}$ for all anticommuting Hermitian matrices $\widehat\alpha_j$, $\widehat\alpha_j^2=\hat I$, $s=0,1$.