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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2401.00784 |
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| _version_ | 1866914870853632000 |
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| author | Brennecke, Christian Brooks, Morris Caraci, Cristina Oldenburg, Jakob |
| author_facet | Brennecke, Christian Brooks, Morris Caraci, Cristina Oldenburg, Jakob |
| contents | We consider dilute Bose gases on the three dimensional unit torus that interact through a pair potential with scattering length of order $ N^{κ-1}$, for some $κ>0$. For the range $ κ\in [0, \frac1{43})$, \cite{ABS} proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $ κ\in [0, \frac1{20})$ by combining some of the key ideas of \cite{ABS} with the novel diagonalization approach introduced recently in \cite{Br}, which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than \cite{ABS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond Brennecke, Christian Brooks, Morris Caraci, Cristina Oldenburg, Jakob Mathematical Physics We consider dilute Bose gases on the three dimensional unit torus that interact through a pair potential with scattering length of order $ N^{κ-1}$, for some $κ>0$. For the range $ κ\in [0, \frac1{43})$, \cite{ABS} proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $ κ\in [0, \frac1{20})$ by combining some of the key ideas of \cite{ABS} with the novel diagonalization approach introduced recently in \cite{Br}, which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than \cite{ABS}. |
| title | A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2401.00784 |