Homogenization of the Schr\"odinger-type equations: operator estimates with correctors

T. A. Suslina

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 93-99

Résumé

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.

Détail

Keywords: periodic differential operators, homogenization, operator error estimates, Schrödinger-type equations.