  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 $i\partial_{\tau} \mathbf{u}_\varepsilon = \mathcal{A}_\varepsilon
\mathbf{u}_\varepsilon + \mathbf{F}$.
