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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.06236 |
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| _version_ | 1866914144048906240 |
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| author | Wu, Zhizhang Zhang, Zhiwen Zhao, Xiaofei |
| author_facet | Wu, Zhizhang Zhang, Zhiwen Zhao, Xiaofei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-Monte Carlo time-splitting methods for Schrödinger equation with Gaussian random potential Wu, Zhizhang Zhang, Zhiwen Zhao, Xiaofei Numerical Analysis 65C30, 65D32, 65M15, 82B44 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. |
| title | Quasi-Monte Carlo time-splitting methods for Schrödinger equation with Gaussian random potential |
| topic | Numerical Analysis 65C30, 65D32, 65M15, 82B44 |
| url | https://arxiv.org/abs/2511.06236 |