Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912469876736000 |
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| author | Brennecke, Christian Kroschinsky, Wilhelm |
| author_facet | Brennecke, Christian Kroschinsky, Wilhelm |
| contents | We revisit the time evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that the system continues to exhibit BEC once the trap has been released and that the dynamics of the condensate is described by the time-dependent Gross-Pitaevskii equation. Like the recent work \cite{BS}, we obtain optimal bounds on the number of excitations orthogonal to the condensate state. In contrast to \cite{BS}, however, whose main strategy consists of controlling the number of excitations with regards to a suitable fluctuation dynamics $t\mapsto e^{-B_t} e^{-iH_Nt}$ with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics $t\mapsto e^{-iH_Nt}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators Brennecke, Christian Kroschinsky, Wilhelm Mathematical Physics Analysis of PDEs We revisit the time evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that the system continues to exhibit BEC once the trap has been released and that the dynamics of the condensate is described by the time-dependent Gross-Pitaevskii equation. Like the recent work \cite{BS}, we obtain optimal bounds on the number of excitations orthogonal to the condensate state. In contrast to \cite{BS}, however, whose main strategy consists of controlling the number of excitations with regards to a suitable fluctuation dynamics $t\mapsto e^{-B_t} e^{-iH_Nt}$ with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics $t\mapsto e^{-iH_Nt}$. |
| title | Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2407.10650 |