On the structure of the lower edge of the spectrum of the periodic magnetic Schrödinger operator with small magnetic potential
Author:
R. G. Shterenberg
Translated by:
the author
Original publication:
Algebra i Analiz, tom 17 (2005), nomer 5.
Journal:
St. Petersburg Math. J. 17 (2006), 865-873
MSC (2000):
Primary 35J10, 35P15
DOI:
https://doi.org/10.1090/S1061-0022-06-00933-2
Published electronically:
July 27, 2006
MathSciNet review:
2241429
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: For the periodic magnetic Schrödinger operator, the structure of the lower edge of the spectrum is investigated. It is known that in the nonmagnetic case the energy depends quadratically on the quasimomentum in the neighborhood of the lower edge of the spectrum. Herewith, the operator admits a convenient “multiplicative” factorization, which makes it possible to investigate the threshold effects efficiently. It is shown that for sufficiently small magnetic potential the magnetic Schrödinger operator also admits a similar factorization.
- M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI https://doi.org/10.1090/S1061-0022-04-00827-1
- Alain Bensoussan and Jens Frehse, Regularity results for nonlinear elliptic systems and applications, Applied Mathematical Sciences, vol. 151, Springer-Verlag, Berlin, 2002. MR 1917320
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭ nye i kvazilineĭ nye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
- J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939), 78–91.
- R. G. Shterenberg, An example of a periodic magnetic Schrödinger operator with a degenerate lower edge of the spectrum, Algebra i Analiz 16 (2004), no. 2, 177–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 2, 417–422. MR 2068347, DOI https://doi.org/10.1090/S1061-0022-05-00858-7
- M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 (Russian). MR 798454
- Giovanni Maria Troianiello, Elliptic differential equations and obstacle problems, The University Series in Mathematics, Plenum Press, New York, 1987. MR 1094820
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Additional Information
R. G. Shterenberg
Affiliation:
Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Email:
roman@RS3759.spb.edu
Keywords:
Periodic operator,
magnetic Schrödinger operator,
lower edge of the spectrum,
threshold effects,
factorization
Received by editor(s):
February 28, 2005
Published electronically:
July 27, 2006
Additional Notes:
Supported by RFBR (grant no. 02-01-00798)
Dedicated:
In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
Article copyright:
© Copyright 2006
American Mathematical Society
