Homogenization of nonstationary Schr\"odinger type equations with periodic coefficients

Abstract

In $L_2(\mathbb{R}^d;{\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/\varepsilon$. We study the behavior of the operator exponential $\exp(-i {\mathcal A}_\varepsilon \tau)$, $\tau \in {\mathbb R}$, for small $\varepsilon$. Approximations for this exponential in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are applied to study the behavior of the solution ${\mathbf u}_\varepsilon$ of the Cauchy problem for the Schr\"odinger type equation $i \partial_\tau {\mathbf u}_\varepsilon = {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon$.