On the spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator $$ \hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3, $$ is proved given that either \(A\in C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)\), 2q > n-2, or the Fourier ser...

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Bibliographic Details
Published inarXiv.org
Main Author Danilov, L I
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 28.05.2009
Subjects
Continuity (mathematics)
Fourier series
Mathematics - Mathematical Physics
Mathematics - Spectral Theory
Operators (mathematics)
Physics - Mathematical Physics
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Summary:The absolute continuity of the spectrum for the periodic Dirac operator $$ \hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3, $$ is proved given that either \(A\in C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)\), 2q > n-2, or the Fourier series of the vector potential \(A:R^n\to R^n\) is absolutely convergent. Here, \(\hat V^{(s)}=(\hat V^{(s)})^*\) are continuous matrix functions and \(\hat V^{(s)}\hat \alpha_j=(-1}^s\hat \alpha_j\hat V^{(s)}\) for all anticommuting Hermitian matrices \(\hat \alpha_j\), \(\hat \alpha_j^2=\hat I\), s=0,1.
ISSN:2331-8422
DOI:10.48550/arxiv.0905.4622