In $L_2 (\mathbb{ R}^d; \mathbb{ C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{ A}_\varepsilon$ with periodic coefficients depending on $\mathbf{ x}/\varepsilon$. We find approximations of the exponential $e^{ -i \tau \mathcal{ A}_\varepsilon}$, $\tau \in \mathbb{ R}$, for small $\varepsilon$ in the ($H^s \to L_2$)-operator norm with suitable $s$. The sharpness of the error estimates with respect to $\tau$ is discussed. The results are applied to study the behavior of the solution $\mathbf{ u}_\varepsilon$ of the Cauchy problem for the Schr\"{ o}dinger-type equation $i\partial_{ \tau} \mathbf{ u}_\varepsilon = \mathcal{ A}_\varepsilon \mathbf{ u}_\varepsilon + \mathbf{ F}$.
Categories: math.AP