  In $L_2({\mathbb R}^d; {\mathbb C}^n)$, we consider a matrix strongly
elliptic differential operator ${A}_\varepsilon$ of order $2p$, $p \geqslant
2$. The operator ${A}_\varepsilon$ is given by ${A}_\varepsilon =
b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, $\varepsilon >0$,
where $g(\mathbf{x})$ is a periodic, bounded, and positive definite
matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential
operator of order $p$. We prove that, for fixed $\tau \in {\mathbb R}$ and
$\varepsilon \to 0$, the operator exponential $e^{-i \tau {A}_\varepsilon}$
converges to $e^{-i \tau {A}^0}$ in the norm of operators acting from the
Sobolev space $H^s({\mathbb R}^d; {\mathbb C}^n)$ (with a suitable $s$) into
$L_2({\mathbb R}^d; {\mathbb C}^n)$. Here $A^0$ is the effective operator.
Sharp-order error estimate is obtained. The results are applied to
homogenization of the Cauchy problem for the Schr\"odinger-type equation $i
\partial_\tau {\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon
+ {\mathbf F}$, ${\mathbf u}_\varepsilon\vert_{\tau=0} = \boldsymbol{\phi}$.
