Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: dependence on time

Abstract

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