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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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  • [BSu1] M. Sh. Birman and T. A. Suslina, Second order periodic differential operators. Threshold properties and homogenization, Algebra i Analiz 15 (2003), no. 5, 1-108; English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639-714. MR 2068790 (2005k:47097)
  • [BSu2] -, Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family, Algebra i Analiz 17 (2005), no. 5, 69-90; English transl., St. Petersburg Math. J. 17 (2006), no. 5, 745-762. MR 2241423 (2008d:47047)
  • [BSu3] -, Homogenization with corrector term for periodic elliptic differential operators, Algebra i Analiz 17 (2005), no. 6, 1-104; English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897-973. MR 2202045 (2006k:35011)
  • [BSu4] -, Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class $ H^1(\mathbb{R}^d)$, Algebra i Analiz 18 (2006), no. 6, 1-130; English transl., St. Petersburg Math. J. 18 (2007), no. 6, 857-955. MR 2307356 (2008d:35008)
  • [K] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., Bd. 132, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [Su1] T. A. Suslina, Homogenization of periodic second order differential operators including first order terms, Spectral Theory of Differential Operators, Amer. Math. Soc. Transl., Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 227-252. MR 2509787 (2010j:35043)
  • [Su2] -, Homogenization in the Sobolev class $ H^1(\mathbb{R}^d)$ for second order periodic elliptic operators with the inclusion of first order terms, Algebra i Analiz 22 (2010), no. 1, 108-222; English transl., St. Petersburg Math. J. 22 (2011), no. 1, 81-162. MR 2641084 (2011d:35041)

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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

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