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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

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
Published electronically: July 27, 2006
MathSciNet review: 2241429
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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.


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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

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00933-2
PII: S 1061-0022(06)00933-2
Keywords: Periodic operator, magnetic Schr\"odinger 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