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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
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.
Received: 07.01.2023
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
Linking options:
https://www.mathnet.ru/eng/aa1870 https://www.mathnet.ru/eng/aa/v35/i3/p138
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Abstract page: | 113 | Full-text PDF : | 3 | References: | 22 | First page: | 17 |
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