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On a class of magnetic Schrödinger operators with discrete spectrum
Author:
N. Anghel
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1613-1616
MSC (2010):
Primary 35J10; Secondary 35P05, 47F05, 81V10
Published electronically:
December 23, 2011
MathSciNet review:
2869145
Full-text PDF
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Additional Information
Abstract: We introduce a class of magnetic Schrödinger operators in  which exhibit pure point spectrum in a fashion that is actually easy to check. This class is an adequate generalization of the more familiar two-dimensional setting, and the proof we give for its spectral discreteness is novel, based on the use of Euclidean Dirac operators coupled to vector potentials.
- [AHS]
J.
Avron, I.
Herbst, and B.
Simon, Schrödinger operators with magnetic fields. I. General
interactions, Duke Math. J. 45 (1978), no. 4,
847–883. MR
518109 (80k:35054)
- [B]
Helga
Baum, Spin-Strukturen und Dirac-Operatoren über
pseudoriemannschen Mannigfaltigkeiten, Teubner-Texte zur Mathematik
[Teubner Texts in Mathematics], vol. 41, BSB B. G. Teubner
Verlagsgesellschaft, Leipzig, 1981 (German). With English, French and
Russian summaries. MR 701244
(84m:58160)
- [D]
Alain
Dufresnoy, Un exemple de champ magnétique dans
𝑅^{𝜈}, Duke Math. J. 50 (1983),
no. 3, 729–734 (French). MR 714827
(84k:78005)
- [HM]
B.
Helffer and A.
Mohamed, Caractérisation du spectre essentiel de
l’opérateur de Schrödinger avec un champ
magnétique, Ann. Inst. Fourier (Grenoble) 38
(1988), no. 2, 95–112 (French, with English summary). MR 949012
(90d:35215)
- [I]
Akira
Iwatsuka, Magnetic Schrödinger operators with compact
resolvent, J. Math. Kyoto Univ. 26 (1986),
no. 3, 357–374. MR 857223
(87j:35287)
- [KS]
Vladimir
Kondratiev and Mikhail
Shubin, Discreteness of spectrum for the magnetic Schrödinger
operators, Comm. Partial Differential Equations 27
(2002), no. 3-4, 477–525. MR 1900553
(2003f:35058), http://dx.doi.org/10.1081/PDE-120002864
- [LM]
H.
Blaine Lawson Jr. and Marie-Louise
Michelsohn, Spin geometry, Princeton Mathematical Series,
vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
(91g:53001)
- [AHS]
- J. Avron, I. Herbst, B. Simon, Schrödinger Operators with Magnetic Fields, I. General Interactions, Duke Math. J.
 , 847-883 (1978). MR 518109 (80k:35054)
- [B]
- H. Baum, Spin-Strukturen und Dirac-Operatoren über Pseudoriemannschen Mannigfaltigkeiten, Teubner Verlagsgesellschaft, Leipzig, 1981. MR 701244 (84m:58160)
- [D]
- A. Dufresnoy, Un Example de Champ Magnétique dans
 , Duke Math. J.  , 729-734 (1983). MR 714827 (84k:78005)
- [HM]
- B. Helffer, A. Mohamed, Caractérisation du Spectre Essentiel de l'Opérateur de Schrödinger avec un Champ Magnétique, Ann. Inst. Fourier
 , 95-112 (1988). MR 949012 (90d:35215)
- [I]
- A. Iwatsuka, Magnetic Schrödinger Operators with Compact Resolvent, J. Math. Kyoto Univ.
 , 357-374 (1986). MR 857223 (87j:35287)
- [KS]
- V. Kondratiev, M. Shubin, Discreteness of Spectrum for the Magnetic Schrödinger Operators, Commun. Partial Diff. Eqns.
 , 477-525 (2002). MR 1900553 (2003f:35058)
- [LM]
- B. Lawson, M-L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
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Additional Information
N. Anghel
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203
Email:
anghel@unt.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11517-X
PII:
S 0002-9939(2011)11517-X
Keywords:
Schrödinger operator,
magnetic field,
discrete spectrum,
Dirac operator
Received by editor(s):
January 5, 2011
Published electronically:
December 23, 2011
Communicated by:
Varghese Mathai
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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