BCS variation method and universal behavior of trapped neutron systems at unitary limit

In this work we study unitary neutron systems confined in a harmonic-oscillator trap, based on the BCS theory and the quasiparticle ring-diagram method. The low-momentum interaction $V_{\text{low-}k}$ renormalized from a fine-tuned CD-Bonn potential with the ${}_{1}S_0$ scattering length approaching the infinity is adopted. The energy ratios of trapped unitary neutron systems with various particle numbers, both those with magic numbers and those with nonmagic ones, to corresponding noninteracting systems, are shown to all flow to a constant value. The BCS wave functions for these trapped unitary systems, with different trap parameters and different particle numbers, manifest themselves with universal patterns: the effect of harmonic-oscillator shells is significant, and moreover, the orbits of the lowest valence shell are occupied with a uniform probability. We also discuss the self-energy given by the BCS calculation, and present a weighted sum rule satisfied by the monopole components of the unitary neutron-neutron interaction. In terms of these, the constant energy ratio independent of the trap parameter and the particle number can be interpreted.