Operator error estimates for homogenization of the nonstationary Schr\"odinger-type equations: dependence on time

TL;DR

This paper develops operator error estimates for the homogenization of nonstationary Schrödinger-type equations with periodic coefficients, analyzing how the approximations of the evolution operator depend on time and small parameters.

Contribution

It provides new operator error estimates for the homogenization of Schrödinger equations, including the dependence on time and the sharpness of these estimates.

Findings

Derived approximations of the exponential operator with small epsilon

Established error bounds in operator norms for the approximations

Discussed the sharpness of error estimates with respect to 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 $e^{-i \tau \mathcal{A}_\varepsilon}$, $\tau \in \mathbb{R}$, for small $\varepsilon$ in the ($H^s \to L_2$)-operator norm with suitable $s$. The sharpness of the error estimates with respect to $\tau$ is discussed. The results are applied to study the behavior of the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for the Schr\"{o}dinger-type equation $i\partial_{\tau} \mathbf{u}_\varepsilon = \mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$.