Unitary limit and linear scaling of neutrons in harmonic trap with tuned CD-Bonn and square-well interactions

We study systems of finite-number neutrons in a harmonic trap at the unitary limit. Two very different types of neutron-neutron interactions are applied, namely, the meson-theoretic CD-Bonn potential and hard-core square-well interactions, all tuned to possess an infinite scattering length, and with effective ranges comparable to the trap size. The potentials are renormalized to equivalent, scattering-length preserving low-momentum potentials, V_low−k, with which the particle-particle hole-hole ring diagrams are summed to all orders to yield the ground-state energy E₀ of the finite neutron system. We find the ratio E₀/E₀ ^free (where E₀ ^free denotes the ground-state energy of the corresponding non-interacting system) to be remarkably independent from variations of the harmonic trap parameter, the type and effective range of the unitarity interaction, and the decimation momentum of V_low−k. Our results support a special linear scaling relation of E₀ versus the trap parameter ħω. Certain properties of Landau's quasi-particles for trapped neutrons at the unitary limit are also discussed.