Threshold approximations for a factorized selfadjoint operator family with the first and second correctors taken into account

Authors:
E. S. Vasilevskaya and T. A. Suslina

Translated by:
T. A. Suslina

Original publication:
Algebra i Analiz, tom **23** (2011), nomer 2.

Journal:
St. Petersburg Math. J. **23** (2012), 275-308

MSC (2010):
Primary 47A55

Published electronically:
January 23, 2012

MathSciNet review:
2841674

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Abstract: In a Hilbert space , a family of operators admitting a factorization of the form , where , , is considered. It is assumed that the point is an isolated eigenvalue of finite multiplicity for . Let be the spectral projection of for the interval (where is sufficiently small). For small , approximations in the operator norm in are obtained for the projection with an error of and for the operator with an error of (the threshold approximations). By using these results, approximation in the operator norm in are constructed for the operator exponential for large with an error of . For the resolvent multiplied by a suitable ``smoothing'' factor, approximation in the operator norm in for small with an error of is obtained. All approximations are given in terms of the spectral characteristics of near the bottom of the spectrum. In these approximations, the first and the second correctors are taken into account. The results are aimed at applications to homogenization problems for periodic differential operators in the small period limit.

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Additional Information

**E. S. Vasilevskaya**

Affiliation:
St. Petersburg State University, Department of Physics, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia

Email:
vasilevskaya-e@yandex.ru

**T. A. Suslina**

Affiliation:
St. Petersburg State University, Department of Physics, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia

Email:
suslina@list.ru

DOI:
https://doi.org/10.1090/S1061-0022-2012-01197-0

Keywords:
Analytic perturbation theory,
threshold approximations,
corrector

Received by editor(s):
June 30, 2010

Published electronically:
January 23, 2012

Additional Notes:
Supported by RFBR (grant no. 08-01-00209-a) and the Program of support of the leading scientific schools (grant NSh-5931.2010.1)

Dedicated:
Dedicated to Vasiliĭ Mikhaĭlovich Babich on the occasion of his birthday

Article copyright:
© Copyright 2012
American Mathematical Society