Homogenization of the periodic Schödinger-type equations

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint strongly elliptic second-order differential operator ${\mathcal A}_\varepsilon$. It is assumed that the coefficients of ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon>0$ is a small parameter. We study the behavior of the operator exponential $e^{-i{\mathcal A}_\varepsilon\tau}$ for small $\varepsilon$ and $\tau \in \mathbb{R}$. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation $i\partial_\tau{\mathbf u}_\varepsilon({\mathbf x},\tau)=({\mathcal A}_\varepsilon{\mathbf u}_\varepsilon)({\mathbf x},\tau)$ with the initial data from a special class.