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 , where the electrical potential contains a Coulomb singularity of arbitrary charge and the magnetic potential is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form where 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.

**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****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****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****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****6.**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****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****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****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-London, 1978. MR**0493421****11.**Upke-Walther Schmincke,*Distinguished selfadjoint extensions of Dirac operators*, Math. Z.**129**(1972), 335–349. MR**0326448****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****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**

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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:
https://doi.org/10.1090/S0002-9947-99-02084-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