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Spectrum of the periodic Dirac operator

Authors:

L I Danilov

Institution:

(1) Physico-Technical Institute, Urals Branch, RAS, Izhevsk, Russia

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 $\hat \alpha _j ,\hat \alpha _j^2 = \hat I,s = 0,1$ . Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 3–17, July, 2000.

Keywords:

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