Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results

Abstract

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 $\mathbf{x}/\epsilon$. We find approximations of the exponential $e^{-i\tau {\mathcal{A}}_{\epsilon }}$, $\tau \in \mathbb{R}$, for small $\epsilon$ 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 ${\mathbf{u}}_{\epsilon }$ of the Cauchy problem for the Schrödinger-type equation $i{\partial }_{\tau }{\mathbf{u}}_{\epsilon }={\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$.

37 pages. arXiv admin note: substantial text overlap with arXiv:1708.00859, arXiv:1606.05868, arXiv:1508.07641