Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients

Abstract

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint strongly elliptic second-order differential operator ${\mathcal A}_\varepsilon$. It is assumed that the coefficients of ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon>0$. We study the behaviour of the operator exponential $e^{-i{\mathcal A}_\varepsilon\tau}$ for small $\varepsilon$ and $\tau \in \mathbb{R}$. The results are applied to the homogenization of solutions of the Cauchy problem for the Schrödinger-type equation $i\partial_\tau{\mathbf u}_\varepsilon({\mathbf x},\tau)=({\mathcal A}_\varepsilon{\mathbf u}_\varepsilon)({\mathbf x},\tau)$ with initial data from a special class. For fixed $\tau$, as $\varepsilon \to 0$, the solution converges in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the solution of the homogenized problem; the error is of the order $O(\varepsilon)$. For fixed $\tau$ we obtain an approximation of the solution ${\mathbf u}_\varepsilon( \cdot ,\tau)$ in the $L_2(\mathbb{R}^d;\mathbb{C}^n)$-norm with error $O(\varepsilon^2)$, and also an approximation of the solution in the $H^1(\mathbb{R}^d;\mathbb{C}^n)$-norm with error $O(\varepsilon)$. In these approximations correctors are taken into account. The dependence of errors on the parameter $\tau$ is traced. Bibliography: 113 items.