The odd fermion at the edge: odd-even staggering in the trapped, unitary Fermi gas

We investigate the odd-even staggering in the harmonically-trapped unitary Fermi gas at large particle-number charge $Q$. Using both a large-$N$ BdG description and a complementary large-charge EFT method, we show that for odd particle number the extra fermion forms an edge-localized quasiparticle near the Thomas-Fermi surface rather than a bulk excitation. In the edge limit, the microscopic BdG problem reduces to a universal coupled Airy system whose lowest positive eigenvalue fixes the leading odd-even splitting energy, $\chi\,\xi^{1/6}(24Q)^{1/9}\,\hbar\omega + \cdots$ where $\xi$ is the Bertsch parameter, and $\chi$ is a universal edge coefficient. The associated EFT describes a fermionic mode confined to the boundary and coupled to the superfluid Goldstone field, reproducing the same $Q$ scaling while introducing a dependence on two low-energy constants. Finally, we numerically compute the spectrum and confirm the predicted scaling and localization properties.