High-energy homogenization of a multidimensional nonstationary Schr\"{o}dinger 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 Schr\"{o}dinger 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.