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 $A_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator $A_\varepsilon$ is given by $A_\varepsilon = b(\mathbf{D})^* g(\frac{\mathbf{x}}{\varepsilon}) b(\mathbf{D})$, $\varepsilon >0$, 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 $\varepsilon \to 0$, the operator exponential $e^{-i \tau A_\varepsilon}$ 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 $i \partial_\tau \mathbf{u}_\varepsilon = A_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$, $\mathbf{u}_\varepsilon\vert_{\tau=0} = \boldsymbol{\phi}$.