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 $i \partial_\tau { \mathbf u}_\varepsilon = { A}_\varepsilon { \mathbf u}_\varepsilon + { \mathbf F}$, ${ \mathbf u}_\varepsilon\vert_{ \tau=0} = \boldsymbol{ \phi}$.
Categories: math.AP