On the spectrum of the periodic Dirac operator

TL;DR

This paper proves the absolute continuity of the spectrum for the periodic Dirac operator under certain regularity conditions on the vector potential, extending understanding of spectral properties in quantum mechanics.

Contribution

It establishes the absolute continuity of the spectrum for the periodic Dirac operator with less restrictive conditions on the vector potential than previously known.

Findings

Spectrum is absolutely continuous under specified regularity conditions.

Results apply to vector potentials with certain Sobolev regularity or absolutely convergent Fourier series.

Advances spectral theory for Dirac operators in mathematical physics.

Abstract

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.