Homogenization of the higher-order Schrödinger-type equations with periodic coefficients

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a matrix strongly elliptic differential operator $A_{\epsilon }$ of order $2p$, $p⩾2$. The operator $A_{\epsilon }$ is given by $A_{\epsilon }=b(\mathbf{D}{)}^{\ast }g(\frac{\mathbf{x}}{\epsilon })b(\mathbf{D})$, $\epsilon >0$, where $g(\mathbf{x})$ is a periodic, bounded, and positive definite matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential operator of order $p$. We prove that, for fixed $\tau \in \mathbb{R}$ and $\epsilon \to 0$, the operator exponential $e^{-i\tau A_{\epsilon }}$ converges to $e^{-i\tau A^0}$ in the norm of operators acting from the Sobolev space $H^s({\mathbb{R}}^d;{\mathbb{C}}^n)$ (with a suitable $s$) into $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$. Here $A^0$ is the effective operator. Sharp-order error estimate is obtained. The results are applied to homogenization of the Cauchy problem for the Schrödinger-type equation $i{\partial }_{\tau }{\mathbf{u}}_{\epsilon }=A_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$, ${\mathbf{u}}_{\epsilon }{|}_{\tau =0}=\mathbf{\varphi }$.