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Approximation of the resolvent of a two-parametric quadratic operator pencil near the bottom of the spectrum


Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 869-891
MSC (2010): Primary 46E30
DOI: https://doi.org/10.1090/S1061-0022-2014-01320-9
Published electronically: July 18, 2014
MathSciNet review: 3184612
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Abstract: A two-parametric pencil of selfadjoint operators $ B(t,\varepsilon )=X(t)^* X(t) + \varepsilon (Y_2^* Y(t)+Y(t)^*Y_2) + \varepsilon ^2 Q$ in a Hilbert space is considered, where $ X(t)=X_0+tX_1$, $ Y(t)=Y_0+tY_1$. It is assumed that the point $ \lambda _0=0$ is an isolated eigenvalue of finite multiplicity for the operator $ X_0^*X_0$, and that the operators $ Y(t)$, $ Y_2$, and $ Q$ are subordinate to $ X(t)$ in a certain sense. The object of study is the generalized resolvent $ (B(t,\varepsilon )+\lambda \varepsilon ^2 Q_0)^{-1}$, where the operator $ Q_0$ is bounded and positive definite. Approximation of this resolvent is obtained for small $ \tau = (t^2 + \varepsilon ^2)^{1/2}$ with an error term of $ O(1)$. This approximation is given in terms of some finite rank operators and is the sum of the principal term and the corrector. The results are aimed at applications to homogenization problems for periodic differential operators in the small period limit.


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Additional Information

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: suslina@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01320-9
Keywords: Analytic perturbation theory, threshold approximations
Received by editor(s): February 3, 2013
Published electronically: July 18, 2014
Additional Notes: Supported by RFBR (grant no. 11-01-00458-a) and the Ministry of education and science of Russian Federation, project 07.09.2012 no. 8501 “2012-1.5-12-000-1003-016”
Article copyright: © Copyright 2014 American Mathematical Society