Operator error estimates for homogenization of the nonstationary Schr\"{o}dinger-type equations: sharpness of the results

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