Geodesic
Homogenization of the Schr\"odinger-type equations: operator estimates with correctors
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 93-99.

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In $L_2(\mathbb R^d;\mathbb C^n)$ we consider a self-adjoint elliptic second-order differential operator $A_\varepsilon$. It is assumed that the coefficients of $A_\varepsilon$ are periodic and depend on $\mathbf x/\varepsilon$, where $\varepsilon>0$ is a small parameter. We study the behavior of the operator exponential $e^{-iA_\varepsilon\tau}$ for small $\varepsilon$ and $\tau\in\mathbb R$. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation $i\partial_\tau \mathbf{u}_\varepsilon(\mathbf x,\tau) = - (A_\varepsilon{\mathbf u}_\varepsilon)(\mathbf x,\tau)$ with initial data in a special class. For fixed $\tau$ and $\varepsilon\to 0$, the solution ${\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)$ converges in $L_2(\mathbb R^d;\mathbb C^n)$ to the solution of the homogenized problem; the error is of order $O(\varepsilon)$. We obtain approximations for the solution ${\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)$ in $L_2(\mathbb R^d;\mathbb C^n)$ with error $O(\varepsilon^2)$ and in $H^1(\mathbb R^d;\mathbb C^n)$ with error $O(\varepsilon)$. These approximations involve appropriate correctors. The dependence of errors on $\tau$ is traced.
Keywords: periodic differential operators, homogenization, operator error estimates, Schrödinger-type equations. 
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     title = {Homogenization of the {Schr\"odinger-type} equations: operator estimates with correctors},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_3_a6/}
}
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T. A. Suslina. Homogenization of the Schr\"odinger-type equations: operator estimates with correctors. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 93-99. http://geodesic.mathdoc.fr/item/FAA_2022_56_3_a6/

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