# A multiscale reduced basis method for Schr\"{o}dinger equation with multiscale and random potentials

## TL;DR

This paper introduces a multiscale reduced basis method for the semiclassical Schrödinger equation with multiscale and random potentials, effectively addressing high-frequency oscillations and enabling efficient numerical simulations in complex quantum systems.

## Contribution

The paper develops a novel multiscale reduced basis approach combining optimization, proper orthogonal decomposition, and quasi-Monte Carlo methods for efficient Schrödinger equation solutions.

## Key findings

- Method reduces spatial grid size proportional to the semiclassical parameter
- Number of random samples inversely proportional to the semiclassical parameter
- Numerical results confirm efficiency and accuracy of the approach

## Abstract

The semiclassical Schr\"{o}dinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schr\"{o}dinger equation with correlated random potentials in both 1D and 2D.