High-frequency homogenization of nonstationary periodic equations
Mark Aleksandrovich Dorodnyi · Applicable Analysis · 2023
In L2(R), we consider an elliptic differential operator Aϵ, ϵ>0, of the form Aϵ=−ddxg(x/ϵ)ddx+ϵ−2V(x/ϵ) with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian Aϵ and for the hyperbolic equation with the operator Aϵ, analogs of homogenization problems, related to the edges of the spectral bands of the operator Aϵ, are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L2(R)-norm for small ε are obtained.