Periodic Homogenization of Schr\"odinger type equations with rapidly oscillating potential

TL;DR

This paper develops a homogenization framework for Schrödinger equations with rapidly oscillating periodic potentials, deriving a macroscopic model using two-scale convergence.

Contribution

It introduces a homogenization approach for Schrödinger equations with time-dependent oscillating potentials, providing a convergence theorem and macroscopic model derivation.

Findings

Proved a convergence theorem for the homogenization process.

Derived the effective macroscopic Schrödinger equation.

Applied two-scale convergence method successfully.

Abstract

This paper is devoted to the homogenization of Shr\"odinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and we derive the macroscopic homogenized model. Our approach is the well known two-scale convergence method.