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An example of a periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum
Author(s):
R.
G.
Shterenberg
Translated by:
the author
Original publication:
Algebra i Analiz,
tom 16
(2004),
vypusk 2.
Journal:
St. Petersburg Math. J.
16
(2005),
417-422.
MSC (2000):
Primary 35J10, 35P15
Posted:
March 9, 2005
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Abstract:
The structure of the lower edge of the spectrum of a periodic magnetic Schrödinger operator is investigated. It is known that in the nonmagnetic case the energy depends quadratically on the quasimomentum in a neighborhood of the lower edge of the spectrum of the operator. An example of a magnetic Schrödinger operator is constructed for which energy is partially degenerate with respect to one component of the quasimomentum.
References:
-
- [BSu]
- M. Sh. Birman and T. A. Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71-107. MR 1882692 (2003f:35220)
- [RSi]
- M. Reed and B. Simon, Methods of modern mathematical physics. 4. Analysis of operators, Academic Press, New York-London, 1978. MR 0493421 (58:12429c)
- [Sk]
- M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 pp.; English transl., Proc. Steklov Inst. Math. 1987, no. 2 (171), 121 pp. MR 0798454 (87h:47110); MR 0905202 (88g:47038)
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Additional Information:
R.
G.
Shterenberg
Affiliation:
St. Petersburg State University, Department of Physics, Petrodvorets, St. Petersburg 198504, Russia
Email:
roman@RS3759.spb.edu
DOI:
10.1090/S1061-0022-05-00858-7
PII:
S 1061-0022(05)00858-7
Keywords:
Periodic operator,
magnetic Schr\"odinger operator,
lower edge of the spectrum,
threshold effects,
degeneration
Received by editor(s):
1/SEP/2003
Posted:
March 9, 2005
Additional Notes:
Supported by RFBR (grant no. 02-01-00798).
Dedicated:
Dedicated to my dear teacher M. Sh. Birman
Copyright of article:
Copyright
2005,
American Mathematical Society
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