Homogenization of nonstationary Schr\"odinger type equations with periodic coefficients

TL;DR

This paper investigates the homogenization of nonstationary Schrödinger equations with periodic coefficients, providing operator exponential approximations and analyzing the behavior of solutions as the period parameter tends to zero.

Contribution

It introduces new operator exponential approximations for nonstationary Schrödinger equations with periodic coefficients, advancing understanding of their asymptotic behavior.

Findings

Operator exponential approximations in Sobolev norms

Effective behavior of Schrödinger solutions in homogenization limit

Quantitative estimates for approximation accuracy

Abstract

In $L_2(\mathbb{R}^d;{\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/\varepsilon$. We study the behavior of the operator exponential $\exp(-i {\mathcal A}_\varepsilon \tau)$, $\tau \in {\mathbb R}$, for small $\varepsilon$. Approximations for this exponential in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are applied to study the behavior of the solution ${\mathbf u}_\varepsilon$ of the Cauchy problem for the Schr\"odinger type equation $i \partial_\tau {\mathbf u}_\varepsilon = {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon$.