Frozen Gaussian approximation for the fractional Schrödinger equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911218588975104 |
|---|---|
| author | Chai, Lihui Chen, Hengzhun Yang, Xu |
| author_facet | Chai, Lihui Chen, Hengzhun Yang, Xu |
| contents | We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schrödinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $δ$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frozen Gaussian approximation for the fractional Schrödinger equation Chai, Lihui Chen, Hengzhun Yang, Xu Numerical Analysis Analysis of PDEs We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schrödinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $δ$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior. |
| title | Frozen Gaussian approximation for the fractional Schrödinger equation |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2403.18287 |