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On homogenization procedure for periodic operators near the edge of an internal gap
Author(s):
M.
Sh.
Birman
Translated by:
T. A. Suslina
Original publication:
Algebra i Analiz,
tom 15
(2003),
vypusk 4.
Journal:
St. Petersburg Math. J.
15
(2004),
507-513.
MSC (2000):
Primary 35P99
Posted:
July 6, 2004
MathSciNet review:
2068979
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Additional information
References:
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- 1.
- A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978, 700 pp. MR 0503330 (82h:35001)
- 2.
- N. S. Bakhvalov and G. P Panasenko, Homogenization of processes in periodic media, ``Nauka'', Moscow, 1984; English transl., Math. Appl. (Soviet. Ser.), vol. 36, Kluwer Acad. Publishers Group, Dordrecht, 1989. MR 0797571 (86m:73049)
- 3.
- V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Averaging of differential operators, ``Nauka'', Moscow, 1993; English transl., Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. MR 1318242 (96h:35003a)
- 4.
- V. V. Zhikov, Spectral approach to asymptotic diffusion problems, Differentsial'nye Uravneniya 25 (1989), no. 1, 44-50; English transl., Differential Equations 25 (1989), no. 1, 33-39. MR 0986395 (90a:35107)
- 5.
- C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math. 57 (1997), no. 6, 1639-1659. MR 1484944 (98j:35017)
- 6.
- M. Birman and T. 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)
- 7.
- M. Sh. Birman, Discrete spectrum in gaps of the perturbed periodic Schrödinger operator. . Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62-89; English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073-1095. MR 1610239 (99h:47054)
- 8.
- M. Sh. Birman and M. Z. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32 (1977), no. 1, 17-84; English transl., Russian Math. Surveys 32 (1977), no. 1, 15-89. MR 0438186 (55:11104)
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Additional Information:
M.
Sh.
Birman
Affiliation:
Department of Physics, St. Petersburg State University, Ul'ynovskaya 1, Petrodvorets, St. Petersburg, 198504, Russia
Email:
tanya@petrov.stoic.spb.su
DOI:
10.1090/S1061-0022-04-00819-2
PII:
S 1061-0022(04)00819-2
Keywords:
Periodic operators,
homogenization,
internal gaps,
threshold effect
Received by editor(s):
2/JUN/2003
Posted:
July 6, 2004
Additional Notes:
Supported by RFBR (grant no.~02-01-00798).
Copyright of article:
Copyright
2004,
American Mathematical Society
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