Abstract
In \(L_2(\mathbb{R}^d)\), we consider a self-adjoint bounded operator \({\mathbb A}_\varepsilon\), \(\varepsilon >0\), of the form
It is assumed that \(a(\mathbf{x})\) is a nonnegative function such that \(a(-\mathbf{x}) = a(\mathbf{x})\) and \(\int_{\mathbb{R}^d} (1+| \mathbf{x} |^4) a(\mathbf{x})\,d\mathbf{x}<\infty\); \(\mu(\mathbf{x},\mathbf{y})\) is \(\mathbb{Z}^d\)-periodic in each variable, \(\mu(\mathbf{x},\mathbf{y}) = \mu(\mathbf{y},\mathbf{x})\) and \(0< \mu_- \leqslant \mu(\mathbf{x},\mathbf{y}) \leqslant \mu_+< \infty\). For small \(\varepsilon\), we obtain an approximation of the resolvent \(({\mathbb A}_\varepsilon + I)^{-1}\) in the operator norm on \(L_2(\mathbb{R}^d)\) with an error of order \(O(\varepsilon^2)\).
DOI 10.1134/S106192084010114
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Funding
The research of A. Piatnitski and E. Zhizhina was partially supported by the project “Pure Mathematics in Norway” and the UiT Aurora project MASCOT. The research of A. Piatnitski was supported by the megagrant of Ministry of Science and Higher Education of the Russian Federation, project 075-15-2022-1115. The research of V. Sloushch and T. Suslina was supported by Russian Science Foundation, project no. 22-11-00092.
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Piatnitski, A., Sloushch, V., Suslina, T. et al. On the Homogenization of Nonlocal Convolution Type Operators. Russ. J. Math. Phys. 31, 137–145 (2024). https://doi.org/10.1134/S106192084010114
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DOI: https://doi.org/10.1134/S106192084010114