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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian


Author: Jingbo Xia
Journal: Trans. Amer. Math. Soc. 351 (1999), 1989-2023
MSC (1991): Primary 47A10, 47F05, 81Q10, 81V45
Published electronically: January 26, 1999
MathSciNet review: 1451618
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the self-adjoint extensions of the Dirac operator $\alpha \cdot (p - B) + \mu _{0}\beta - W$, where the electrical potential $W$ contains a Coulomb singularity of arbitrary charge and the magnetic potential $B$ is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form $v(r)/r$ where $v$ has a limit at 0, then, for any self-adjoint extension of the Dirac operator, removing the singularity results in a Hilbert-Schmidt perturbation of its resolvent.


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  • 1. C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, 1988.
  • 2. Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. MR 928802 (89e:46001)
  • 3. I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142 (39 #7447)
  • 4. Hubert Kalf, A limit-point criterion for separated Dirac operators and a little known result on Riccati’s equation, Math. Z. 129 (1972), 75–82. MR 0315196 (47 #3745)
  • 5. G. Chavent and P. Lemonnier, Identification de la non-linéarité d’une équation parabolique quasilinéaire, Appl. Math. Optim. 1 (1974/75), no. 2, 121–162. MR 0397171 (53 #1031)
  • 6. Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617 (53 #11389)
  • 7. M. A. Naĭmark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. MR 0262880 (41 #7485)
  • 8. G. Nenciu, Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms, Comm. Math. Phys. 48 (1976), no. 3, 235–247. MR 0421456 (54 #9459)
  • 9. M. Rose, Relativistic electron theory, Wiley, New York, 1961.
  • 10. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. MR 0493421 (58 #12429c)
  • 11. Upke-Walther Schmincke, Distinguished selfadjoint extensions of Dirac operators, Math. Z. 129 (1972), 335–349. MR 0326448 (48 #4792)
  • 12. E. C. Titchmarsh, On the nature of the spectrum in problems of relativistic quantum mechanics. III, Quart. J. Math. Oxford Ser. (2) 13 (1962), 255–263. MR 0147697 (26 #5211)
  • 13. Rainer Wüst, Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials, Math. Z. 141 (1975), 93–98. MR 0365233 (51 #1486)

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Additional Information

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214
Email: jxia@acsu.buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02084-X
PII: S 0002-9947(99)02084-X
Keywords: Dirac operator, spectrum, perturbation
Received by editor(s): January 15, 1996
Received by editor(s) in revised form: February 27, 1997
Published electronically: January 26, 1999
Additional Notes: Research supported in part by National Science Foundation grant DMS-9400600.
Article copyright: © Copyright 1999 American Mathematical Society