Homogenization of the higher-order Schrödinger-type equations with periodic coefficients
Tatiana Aleksandrovna Suslina · EMS series of congress reports · 2021
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 $i \partial\_\tau \mathbf{u}\varepsilon = A\varepsilon \mathbf{u}\varepsilon + \mathbf{F}$, $\mathbf{u}\varepsilon\vert\_{\tau=0} = \boldsymbol{\phi}$.