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 $i \partial_\tau { \mathbf u}_\varepsilon = { \mathcal A}_\varepsilon { \mathbf u}_\varepsilon$.
Categories: math.AP