Localization for the Schr\"{o}dinger equation in a locally periodic medium

TL;DR

This paper investigates the homogenization of the Schr"{o}dinger equation in a locally periodic medium, demonstrating the existence of localized solutions that combine Bloch waves with an effective quadratic potential Schr"{o}dinger equation.

Contribution

It introduces a homogenization framework for Schr"{o}dinger equations in locally periodic media with semi-classical scaling and well-prepared initial data, revealing localized asymptotic solutions.

Findings

Existence of localized solutions combining Bloch waves and homogenized Schr"{o}dinger solutions.

Asymptotic behavior characterized by a product of Bloch eigenfunctions and quadratic potential solutions.

Validation of homogenization approach for semi-classical Schr"{o}dinger equations in complex media.

Abstract

We study the homogenization of a Schr\"{o}dinger equation in a locally periodic medium. For the time and space scaling of semi-classical analysis we consider well-prepared initial data that are concentrated near a stationary point (with respect to both space and phase) of the energy, i.e. the Bloch cell eigenvalue. We show that there exists a localized solution which is asymptotically given as the product of a Bloch wave and of the solution of an homogenized Schr\"{o}dinger equation with quadratic potential.