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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Tadashi Ikuta and Kazuhisa Shima
Journal: Proc. Amer. Math. Soc. 131 (2003), 1471-1479
MSC (1991): Primary 34L05, 34L40
Published electronically: September 20, 2002
MathSciNet review: 1949877
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Abstract: 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.


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Additional Information

Tadashi Ikuta
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278-8510, Japan
Email: ikuta_tadashi@ma.noda.tus.ac.jp

Kazuhisa Shima
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278-8510, Japan
Email: shima@rs.noda.tus.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06661-3
PII: S 0002-9939(02)06661-3
Keywords: Locally compact operator, Dirac operator, essential spectrum, discrete spectrum
Received by editor(s): November 29, 2000
Received by editor(s) in revised form: March 29, 2001, and December 11, 2001
Published electronically: September 20, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society