The frequencies of the analytic normal modes for $N$ identical particles are studied as a function of system parameters from the weakly interacting BCS regime to the strongly interacting unitary regime. The normal modes were obtained previously from a first-order $L=0$ group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction for confined identical particles. In a precursor to this study, the collective behavior of these normal modes was investigated as a function of $N$ from few-body systems to many-body systems analyzing the contribution of individual particles to the collective macroscopic motions. A specific case, the Hamiltonian for Fermi gases in the unitary regime, was studied in more detail. This regime is known to support collective behavior in the form of superfluidity and has previously been successfully described using normal modes. Two phenomena that could sustain the emergence and stability of superfluid behavior were revealed, including the behavior of the normal mode frequencies as $N$ increases. In this paper, I focus on a more detailed analysis of these analytic frequencies, extending my investigation to include Hamiltonians with a range of interparticle interaction strengths from the BCS regime to the unitary regime and analyzing the microscopic dynamics that lead to large gaps at unitarity. The results of the present study suggest that in regimes where higher-order effects are small, normal modes can be used to describe the physics of superfluidity from the weakly interacting BCS regime with the emergence of small excitation gaps to unitarity with its large gaps and can offer insight into a possible microscopic understanding of the behavior at unitarity. This approach could thus offer an alternative to the two-body pairing models commonly used to describe superfluidity along this transition.

中文翻译：

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相同粒子的解析正常模式频率：从 BCS 到单一性激发间隙出现和稳定性的微观动力学

解析正常模式的频率为 $N$相同的粒子作为系统参数的函数进行研究，从弱相互作用的 BCS 机制到强相互作用的单一机制。正常模式是先前从一阶获得的$升=0$具有约束相同粒子的一般二体相互作用的三维哈密顿量的群论解。在本研究的先导中，研究了这些正常模式的集体行为作为$N$从少体系统到多体系统，分析单个粒子对集体宏观运动的贡献。更详细地研究了一个特殊情况，即统一制度中费米气体的哈密顿量。众所周知，这种机制以超流动的形式支持集体行为，并且之前已经使用正常模式成功地描述过。揭示了可以维持超流体行为的出现和稳定性的两种现象，包括正常模式频率的行为作为$N$增加。在本文中，我专注于对这些解析频率进行更详细的分析，将我的研究扩展到包括具有从 BCS 机制到单一机制的一系列粒子间相互作用强度的哈密顿量，并分析导致单一性大间隙的微观动力学。本研究的结果表明，在高阶效应很小的情况下，正常模式可用于描述从弱相互作用 BCS 机制出现的小激发间隙到具有大间隙的单一性的超流物理，并且可以提供对统一行为可能的微观理解的洞察。因此，这种方法可以为通常用于描述沿此过渡的超流动性的双体配对模型提供替代方案。