Homogenization of Schrödinger-Type equations

We consider a self-adjoint elliptic operator $A_\varepsilon$, $\varepsilon >0$, on $L_2({\mathbb R}^d;{\mathbb C}^n)$ given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D})$. Here $b({\mathbf D})=\sum_{j=1}^d b_j D_j$ is a first-order matrix differential operator such that the symbol $b(\boldsymbol{\xi})$ has maximal rank. The matrix-valued function $g({\mathbf x})$ is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential $e^{- i \tau A_\varepsilon}$, where $\tau \in {\mathbb R}$. It is shown that, as $\varepsilon \to 0$, the operator $e^{- i \tau A_\varepsilon}$ 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 suitable $s$) to $L_2({\mathbb R}^d;{\mathbb C}^n)$. Here $A^0$ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation $i \partial_\tau {\mathbf u}_\varepsilon ({\mathbf x}, \tau)=A_\varepsilon {\mathbf u}_\varepsilon({\mathbf x}, \tau)$.