An approach to the spectrum structure of Dirac operators by the localcompactness method
Authors:
Tadashi Ikuta and Kazuhisa Shima
Journal:
Proc. Amer. Math. Soc. 131 (2003), 14711479
MSC (1991):
Primary 34L05, 34L40
Published electronically:
September 20, 2002
MathSciNet review:
1949877
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.
 1.
M. Arai, On essential SelfAdjointness of Dirac Operators, RIMS Kokyuroku 242 (1975), 1021.
 2.
M. Arai and O. Yamada, Essential Selfadjointness and Invariance of the Essential Spectrum for Dirac Operators, Publ. RIMS 18 (1982), 973985.
 3.
John
B. Conway, A course in functional analysis, 2nd ed., Graduate
Texts in Mathematics, vol. 96, SpringerVerlag, New York, 1990. MR 1070713
(91e:46001)
 4.
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, SpringerVerlag, Berlin, 1987. MR 883643
(88g:35003)
 5.
P.
Deift, W.
Hunziker, B.
Simon, and E.
Vock, Pointwise bounds on eigenfunctions and wave packets in
𝑁body quantum systems. IV, Comm. Math. Phys.
64 (1978/79), no. 1, 1–34. MR 516993
(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, 769.
 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 (Univ. Dundee,
Dundee, 1972) Springer, Berlin, 1972, pp. 87–102. Lecture
Notes in Math., Vol. 280. MR 0420030
(54 #8047)
 9.
Tosio
Kato, Perturbation theory for linear operators, Classics in
Mathematics, SpringerVerlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
(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.
Peter
A. Perry, Scattering theory by the Enss method, Mathematical
Reports, vol. 1, Harwood Academic Publishers, Chur, 1983. Edited by B.
Simon. MR
752694 (85k:35181)
 12.
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. I. Functional
analysis, Academic Press, New YorkLondon, 1972. MR 0493419
(58 #12429a)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. II. Fourier
analysis, selfadjointness, Academic Press [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1975. MR 0493420
(58 #12429b)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. III, Academic
Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1979.
Scattering theory. MR 529429
(80m:81085)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. IV. Analysis of
operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1978. MR 0493421
(58 #12429c)
 13.
Barry
Simon, Trace ideals and their applications, London
Mathematical Society Lecture Note Series, vol. 35, Cambridge
University Press, CambridgeNew York, 1979. MR 541149
(80k:47048)
 14.
Bernd
Thaller, The Dirac equation, Texts and Monographs in Physics,
SpringerVerlag, Berlin, 1992. MR 1219537
(94k:81056)
 15.
Joachim
Weidmann, Linear operators in Hilbert spaces, Graduate Texts
in Mathematics, vol. 68, SpringerVerlag, New YorkBerlin, 1980.
Translated from the German by Joseph Szücs. MR 566954
(81e:47001)
 16.
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
(98e:35122), http://dx.doi.org/10.14492/hokmj/1351257976
 17.
G. Zhislin, Discussion of the Spectrum of the Schrödinger Operator for Systems of Several Particles, Tr. Mosk. Mat. Obs. 9 (1960), 81128.
 1.
 M. Arai, On essential SelfAdjointness of Dirac Operators, RIMS Kokyuroku 242 (1975), 1021.
 2.
 M. Arai and O. Yamada, Essential Selfadjointness and Invariance of the Essential Spectrum for Dirac Operators, Publ. RIMS 18 (1982), 973985.
 3.
 J. B. Conway, A Course in Functional Analysis, 2nd ed., SpringerVerlag, 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, SpringerVerlag, Berlin, 1987. MR 88g:35003
 5.
 P. Deift, W. Hunziker, B. Simon, and E. Vock, Pointwise Bounds on Eigenfunctions and Wave Packets in Body Quantum Systems IV, Commun. Math. Phys. 64 (1978), 134. 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, 769.
 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, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1980. MR 54:8047
 9.
 T. Kato, Perturbation Theory for Linear Operators, Corrected Printing of 2nd ed., SpringerVerlag, 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. IIV, Academic Press, New York, 19721979. 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, SpringerVerlag, BerlinHeidelbergNew York, 1992. MR 94k:81056
 15.
 J. Weidmann, Linear Operators in Hilbert Spaces, English ed., SpringerVerlag, 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), 439449. 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), 81128.
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Additional Information
Tadashi Ikuta
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 2788510, 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 2788510, Japan
Email:
shima@rs.noda.tus.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993902066613
PII:
S 00029939(02)066613
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. TomczakJaegermann
Article copyright:
© Copyright 2002
American Mathematical Society
