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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
DOI: https://doi.org/10.1090/S1061-0022-2012-01197-0
Published electronically: January 23, 2012
MathSciNet review: 2841674
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Abstract: In a Hilbert space $ \mathfrak{H}$, a family of operators $ A(t)$ admitting a factorization of the form $ A(t)= X(t)^*X(t)$, where $ X(t)=X_0 +tX_1$, $ t \in \mathbb{R}$, is considered. It is assumed that the point $ \lambda _0=0$ is an isolated eigenvalue of finite multiplicity for $ A(0)$. Let $ F(t)$ be the spectral projection of $ A(t)$ for the interval $ [0,\delta ]$ (where $ \delta $ is sufficiently small). For small $ \vert t\vert$, approximations in the operator norm in $ \mathfrak{H}$ are obtained for the projection $ F(t)$ with an error of $ O(\vert t\vert^3)$ and for the operator $ A(t)F(t)$ with an error of $ O(\vert t\vert^5)$ (the threshold approximations). By using these results, approximation in the operator norm in $ \mathfrak{H}$ are constructed for the operator exponential $ \exp (-A(t)\tau )$ for large $ \tau >0$ with an error of $ O(\tau ^{-3/2})$. For the resolvent $ (A(t)+\varepsilon ^2 I)^{-1}$ multiplied by a suitable ``smoothing'' factor, approximation in the operator norm in $ \mathfrak{H}$ for small $ \varepsilon >0$ with an error of $ O(\varepsilon )$ is obtained. All approximations are given in terms of the spectral characteristics of $ A(t)$ 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

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