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

TL;DR

This paper develops a high-energy homogenization approach for a multidimensional nonstationary Schrödinger equation with periodic coefficients, providing approximations for solutions in the high-energy regime as the parameter epsilon approaches zero.

Contribution

It introduces a novel homogenization method for the nonstationary Schrödinger equation at arbitrary points of the dispersion relation, extending high-energy analysis to multidimensional settings.

Findings

Derived $L_2$-norm approximations for solutions with small epsilon

Extended homogenization techniques to nonstationary Schrödinger equations

Analyzed behavior at arbitrary points of the dispersion relation

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.