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arXiv · 2511.06236

Quasi-Monte Carlo time-splitting methods for the Schr\"odinger equation with Gaussian random potential

Zhizhang Wu·Zhiwen Zhang·Xiaofei Zhao
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Abstract

In this paper, we study the Schr\"odinger 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.

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Zhizhang Wu, Zhiwen Zhang, Xiaofei Zhao. 2025-11-09. Quasi-Monte Carlo time-splitting methods for the Schr\"odinger equation with Gaussian random potential. https://arxiv.org/abs/2511.06236

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