arXiv 2401.007842024-01math-phmath.MP

Gross-Pitaevskii 区域及更广泛情形下玻色-爱因斯坦凝聚的一个简短证明

A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond

Christian Brennecke, Morris Brooks, Cristina Caraci, Jakob Oldenburg

备注：: 13 pages + one appendix; accepted journal version
发表信息：: Ann. Henri Poincare (2024)
DOI：: 10.1007/s00023-024-01465-8

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AI总结：

该研究结合已有核心思想与基于Schur补公式的新型对角化方法，给出了三维稀薄玻色气体在更广参数范围内基态玻色-爱因斯坦凝聚的自包含证明，避免了算符指数的使用，大幅简化了证明过程。

摘要：

我们研究三维单位环面上的稀薄玻色气体，其相互作用对势的散射长度量级为 $N^{κ-1}$，其中 $κ>0$。对于 $κ ∈ [0, \frac1{43})$ 的范围，文献\cite{ABS}基于产生和湮灭算符的四次方算符指数幺正重整化，证明了低能态到零动量模的完全玻色-爱因斯坦凝聚（BEC）。本文结合文献\cite{ABS}的部分核心思想与近期文献\cite{Br}提出的基于Schur补公式的新型对角化方法，给出了 $κ ∈ [0, \frac1{20})$ 时基态BEC的全新自包含证明。特别地，我们的证明无需使用算符指数，且比文献\cite{ABS}的证明简洁得多。

英文摘要：

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}.

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