Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908393679093760 |
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| author | Kammerer, Clotilde Fermanian Lasser, Caroline Robert, Didier |
| author_facet | Kammerer, Clotilde Fermanian Lasser, Caroline Robert, Didier |
| contents | This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schrödinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schrödinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_17053 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings Kammerer, Clotilde Fermanian Lasser, Caroline Robert, Didier Analysis of PDEs Mathematical Physics This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schrödinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schrödinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter. |
| title | Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2305.17053 |