  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 current study suggest that in regimes where higher-order
effects are small, normal modes an 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.
