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Algebra i Analiz, 2023, Volume 35, Issue 3, Pages 138–184 (Mi aa1870)  

This article is cited in 3 scientific papers (total in 3 papers)

Research Papers

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

T. A. Suslina

Saint Petersburg State University
References:
Abstract: In a Hilbert space $\mathfrak H$, we consider a family of selfadjoint operators (a quadratic operator pencil) $A(t)$, $t\in\mathbb{R}$, of the form $A(t) = X(t)^* X(t)$, where $X(t) = X_0 + t X_1$. It is assumed that the point $\lambda_0=0$ is an isolated eigenvalue of finite multiplicity for the operator $A(0)$. Let $F(t)$ be the spectral projection of the operator $A(t)$ for the interval $[0,\delta]$. Using approximations for $F(t)$ and $A(t)F(t)$ for $|t| \leqslant t_0$ (the so-called threshold approximations), we obtain approximations in the operator norm on $\mathfrak H$ for the operator exponential $\exp(-i \tau A(t))$, $\tau \in \mathbb{R}$. The numbers $\delta$ and $t_0$ are controlled explicitly. Next, we study the behavior for small $\varepsilon >0$ of the operator $\exp(-i \varepsilon^{-2} \tau A(t))$ multiplied by the “smoothing factor” $\varepsilon^s (t^2 + \varepsilon^2)^{-s/2}$ with a suitable $s>0$. The obtained approximations are given in terms of the spectral characteristics of the operator $A(t)$ near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.
Keywords: homogenization, quadratic operator pencils, operator exponential, threshold approximations, analytic perturbation theory.
Funding agency Grant number
Russian Science Foundation 22-11-00092
Received: 07.01.2023
Document Type: Article
Language: Russian
Citation: T. A. Suslina, “Threshold approximations for the exponential of a factorized operator family with correctors taken into account”, Algebra i Analiz, 35:3 (2023), 138–184
Citation in format AMSBIB
\Bibitem{Sus23}
\by T.~A.~Suslina
\paper Threshold approximations for the exponential of a factorized operator family with correctors taken into account
\jour Algebra i Analiz
\yr 2023
\vol 35
\issue 3
\pages 138--184
\mathnet{http://mi.mathnet.ru/aa1870}
Linking options:
  • https://www.mathnet.ru/eng/aa1870
  • https://www.mathnet.ru/eng/aa/v35/i3/p138
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и анализ St. Petersburg Mathematical Journal
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    Abstract page:113
    Full-text PDF :3
    References:22
    First page:17
     
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