  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 infinite scattering lengths, and with
effective ranges comparable to or larger than the trap size. The potentials are
renormalized to equivalent, scattering-length preserving low-momentum
potentials, $V_{{\rm low}-k}$, with which the particle-particle hole-hole ring
diagrams are summed to all orders to yield the ground-state energy $E_0$ of the
finite neutron system. We find the ratio $E_0/E_0^{\rm free}$ (where $E_0^{\rm
free}$ denotes the ground-state energy of the corresponding non-interacting
system) to be remarkably independent from variations of the harmonic trap
parameter, the number of neutrons, the decimation momentum of $V_{{\rm
low}-k}$, and the type and effective range of the unitarity potential. Our
results support a special virial linear scaling relation of $E_0$. Certain
properties of Landau's quasi-particles for trapped neutrons at the unitary
limit are also discussed.
