Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Arai, On essential Self-Adjointness of Dirac Operators, RIMS Kokyuroku 242 (1975), 10-21.
  • 2. M. Arai and O. Yamada, Essential Self-adjointness and Invariance of the Essential Spectrum for Dirac Operators, Publ. RIMS 18 (1982), 973-985.
  • 3. J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer-Verlag, New York, Tokyo, 1990. MR 91e:46001
  • 4. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, 1987. MR 88g:35003
  • 5. P. Deift, W. Hunziker, B. Simon, and E. Vock, Pointwise Bounds on Eigenfunctions and Wave Packets in $N$-Body Quantum Systems IV, Commun. Math. Phys. 64 (1978), 1-34. MR 80k:81016
  • 6. V. Enss, Geometrical Methods in Spectral and Scattering Theory of Schrödinger Operators, in Rigorous Atomic and Molecular Physics edited by G. Velo and A. S. Wightman, Plenum, New York, 1981, 7-69.
  • 7. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, With Application to Schrödinger Operators, Applied Mathematical Sciences, vol. 113, Springer, New York, 1996.
  • 8. K. Jörgens, Perturbations of the Dirac Operator. Conference on the theory of ordinary and partial differential equations, Lecture Note in Mathematics, vol. 280, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1980. MR 54:8047
  • 9. T. Kato, Perturbation Theory for Linear Operators, Corrected Printing of 2nd ed., Springer-Verlag, Berlin, 1980. MR 96a:47025
  • 10. S. Nakamura, Lectures on Schrödinger operators, Lectures in Mathematical Sciences, vol. 6, Graduate School of Mathematical Sciences, University of Tokyo, 1994.
  • 11. P. A. Perry, Scattering Theory by the Enss Method, Mathematical Reports, vol. 1, Harwood, New York, 1983. MR 85k:35181
  • 12. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. I-IV, Academic Press, New York, 1972-1979. MR 58:12429a; MR 58:12429b; MR 80m:81085; MR 58:12429c
  • 13. B. Simon, Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, 1979. MR 80k:47048
  • 14. B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg-New York, 1992. MR 94k:81056
  • 15. J. Weidmann, Linear Operators in Hilbert Spaces, English ed., Springer-Verlag, Berlin, 1980. MR 81e:47001
  • 16. O. Yamada, On the Spectrum of Dirac Operators with the Unbounded Potential at Infinity, Hokkaido Math. J. 26 (1997), 439-449. MR 98e:35122
  • 17. G. Zhislin, Discussion of the Spectrum of the Schrödinger Operator for Systems of Several Particles, Tr. Mosk. Mat. Obs. 9 (1960), 81-128.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34L05, 34L40

Retrieve articles in all journals with MSC (1991): 34L05, 34L40

Additional Information

Tadashi Ikuta
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278-8510, Japan

Kazuhisa Shima
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278-8510, Japan

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

American Mathematical Society