FEM-DtN-SIM Method for Computing Resonances of Schrödinger Operators
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917986678341632 |
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| author | Gong, Bo Sato, Takumi Sun, Jiguang Wu, Xinming |
| author_facet | Gong, Bo Sato, Takumi Sun, Jiguang Wu, Xinming |
| contents | The study of resonances of the Schrödinger operator has a long-standing tradition in mathematical physics. Extensive theoretical investigations have explored the proximity of resonances to the real axis, their distribution, and bounds on the counting functions. However, computational results beyond one dimension remain scarce due to the nonlinearity of the problem and the unbounded nature of the domain. We propose a novel approach that integrates finite elements, Dirichlet-to-Neumann (DtN) mapping, and the spectral indicator method. The DtN mapping, imposed on the boundary of a truncated computational domain, enforces the outgoing condition. Finite elements allow for the efficient handling of complicated potential functions. The spectral indicator method effectively computes (complex) eigenvalues of the resulting nonlinear algebraic system without introducing spectral pollution. The viability of this approach is demonstrated through a range of numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | FEM-DtN-SIM Method for Computing Resonances of Schrödinger Operators Gong, Bo Sato, Takumi Sun, Jiguang Wu, Xinming Numerical Analysis 78M10, 65H17, 65N25 The study of resonances of the Schrödinger operator has a long-standing tradition in mathematical physics. Extensive theoretical investigations have explored the proximity of resonances to the real axis, their distribution, and bounds on the counting functions. However, computational results beyond one dimension remain scarce due to the nonlinearity of the problem and the unbounded nature of the domain. We propose a novel approach that integrates finite elements, Dirichlet-to-Neumann (DtN) mapping, and the spectral indicator method. The DtN mapping, imposed on the boundary of a truncated computational domain, enforces the outgoing condition. Finite elements allow for the efficient handling of complicated potential functions. The spectral indicator method effectively computes (complex) eigenvalues of the resulting nonlinear algebraic system without introducing spectral pollution. The viability of this approach is demonstrated through a range of numerical examples. |
| title | FEM-DtN-SIM Method for Computing Resonances of Schrödinger Operators |
| topic | Numerical Analysis 78M10, 65H17, 65N25 |
| url | https://arxiv.org/abs/2504.11680 |