Quasi-Monte Carlo time-splitting methods for the Schrödinger equation with Gaussian random potential

2025/11/09 by Wu, Zhizhang, Zhang, Zhiwen, Zhao, Xiaofei
Decision Sciences · Mathematics · #65C30 #65D32 #65M15 #82B44 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design

paper · doi:10.48550/arxiv.2511.06236

openalex publication_date 2025/11/09 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.

Citations

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