High-energy homogenization of a multidimensional nonstationary Schrödinger equation

Abstract

In $L_2({\mathbb{R}}^d)$, we consider an elliptic differential operator ${\mathcal{A}}_{\epsilon }=-\mathrm{div}g(\mathbf{x}/\epsilon )\nabla +{\epsilon }^{-2}V(\mathbf{x}/\epsilon )$, $\epsilon >0$, with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian ${\mathcal{A}}_{\epsilon }$, analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator ${\mathcal{A}}_1$ are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in $L_2({\mathbb{R}}^d)$-norm for small $\epsilon$ are obtained.

26 pages