An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator $H=H_0+V=-ic\alpha\cdot\nabla+\beta mc^2+V$. The local compactness of $H$ is shown under some assumption on $V$. This method enables us to prove that if $\vert V(x)-a\beta\vert\to 0$ as $\vert x\vert\to\infty$, then $\sigma_{\operatorname{ess}}(H)=(-\infty,-mc^2-a]\cup [mc^2+a,\infty)$ and to give a significant sufficient condition that $H^{+}$ or $H^{-}$ has a purely discrete spectrum.