High-frequency homogenization of nonstationary periodic equations

Abstract

We consider an elliptic differential operator $A_{\epsilon }=-\frac{d}{dx}g(x/\epsilon )\frac{d}{dx}+{\epsilon }^{-2}V(x/\epsilon )$, $\epsilon >0$, with periodic coefficients acting in $L_2(\mathbb{R})$. For the nonstationary Schrödinger equation with the Hamiltonian $A_{\epsilon }$ and for the hyperbolic equation with the operator $A_{\epsilon }$, analogs of homogenization problems, related to the edges of the spectral bands of the operator $A_{\epsilon }$, are studied (the so called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in $L_2(\mathbb{R})$-norm for small $\epsilon$ are obtained.

25 pages