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

Ikuta, Tadashi; Shima, Kazuhisa

doi:10.1090/S0002-9939-02-06661-3

An approach to the spectrum structure of Dirac operators by the local-compactness method
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by Tadashi Ikuta and Kazuhisa Shima PDF

Proc. Amer. Math. Soc. 131 (2003), 1471-1479 Request permission

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 $|V(x)-a\beta |\to 0$ as $|x|\to \infty$, 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

M. Arai, On essential Self-Adjointness of Dirac Operators, RIMS Kokyuroku 242 (1975), 10–21.
M. Arai and O. Yamada, Essential Self-adjointness and Invariance of the Essential Spectrum for Dirac Operators, Publ. RIMS 18 (1982), 973–985.
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643, DOI 10.1007/978-3-540-77522-5
P. Deift, W. Hunziker, B. Simon, and E. Vock, Pointwise bounds on eigenfunctions and wave packets in $N$-body quantum systems. IV, Comm. Math. Phys. 64 (1978/79), no. 1, 1–34. MR 516993, DOI 10.1007/BF01940758
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.
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.
K. Jörgens, Perturbations of the Dirac operator, Conference on the Theory of Ordinary and Partial Differential Equations (Univ. Dundee, Dundee, 1972) Lecture Notes in Math., Vol. 280, Springer, Berlin, 1972, pp. 87–102. MR 0420030
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
S. Nakamura, Lectures on Schrödinger operators, Lectures in Mathematical Sciences, vol. 6, Graduate School of Mathematical Sciences, University of Tokyo, 1994.
Peter A. Perry, Scattering theory by the Enss method, Mathematical Reports, vol. 1, Harwood Academic Publishers, Chur, 1983. Edited by B. Simon. MR 752694
Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149, DOI 10.1007/BFb0064579
Bernd Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. MR 1219537, DOI 10.1007/978-3-662-02753-0
Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954, DOI 10.1007/978-1-4612-6027-1
Osanobu Yamada, On the spectrum of Dirac operators with the unbounded potential at infinity, Hokkaido Math. J. 26 (1997), no. 2, 439–449. MR 1463096, DOI 10.14492/hokmj/1351257976
G. Zhislin, Discussion of the Spectrum of the Schrödinger Operator for Systems of Several Particles, Tr. Mosk. Mat. Obs. 9 (1960), 81–128.