Homogenization of nonstationary Schrödinger type equations with periodic coefficients

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal{A}}_{\epsilon }$ with periodic coefficients depending on $\mathbf{x}/\epsilon$. We study the behavior of the operator exponential $\exp (-i{\mathcal{A}}_{\epsilon }\tau )$, $\tau \in \mathbb{R}$, for small $\epsilon$. 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 ${\mathbf{u}}_{\epsilon }$ of the Cauchy problem for the Schrödinger type equation $i{\partial }_{\tau }{\mathbf{u}}_{\epsilon }={\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }$.

arXiv comment78 pages