A model reduction method for solving the eigenvalue problem of semiclassical random Schr\"odinger operators

In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schr\"odinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce the multiscale finite element method (MsFEM) to approximate the resulting parametric EVP. We then use the quasi-Monte Carlo (qMC) method to calculate empirical statistics within a finite-dimensional random space. Furthermore, using a set of low-dimensional proper orthogonal decomposition (POD) basis functions, the referred degrees of freedoms for constructing multiscale basis are independent of the spatial mesh. Given the bounded assumption on the random potentials, we then derive and prove an error estimate for the proposed method. Finally, we conduct numerical experiments to validate the error estimate. In addition, we investigate the localization of eigenfunctions for the Schr\"odinger operator with spatially random potentials. The results show that our method provides a practical and efficient solution for simulating complex quantum systems governed by semiclassical random Schr\"odinger operators.