High-frequency homogenization of nonstationary periodic equations

Abstract

ABSTRACT In L 2 ( R ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − d d x g ( x / ϵ ) d d x + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian A ϵ and for the hyperbolic equation with the operator A ϵ , analogs of homogenization problems, related to the edges of the spectral bands of the operator A ϵ , 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 ( R ) -norm for small ε are obtained.