High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation

Abstract

In \(L_2(\mathbb{R}^d)\), we consider an elliptic differential operator \(\mathcal{A}_\varepsilon \! = \! - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)\), \( \varepsilon > 0\), with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian \(\mathcal{A}_\varepsilon\), 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 \(\varepsilon\) are obtained.

DOI 10.1134/S1061920823040064