Surface corrections and vortex energetics in the Thomas--Fermi limit of a self-bound unitary Fermi superfluid

We study a spin-balanced unitary Fermi superfluid bound by an attractive $1\mathrm{/}r$ interaction within an extended Thomas–Fermi density functional. Its large-particle-number limit is the Lane–Emden polytrope of index $3\mathrm{/}2$. We prove the leading gradient correction to the ground-state energy and strong convergence of the density amplitude. A rescaled surface problem predicts the next energy correction and the leading integrated density error. Matching a vortex core to the polytropic bulk gives the energetic threshold of a singly quantized central vortex, including the constant beyond the leading logarithm. Linearizing the same surface problem predicts a local energetic threshold for collective surface excitations that, in the natural TF frequency units, decreases more slowly with particle number than the central-vortex threshold. Radial stationary, spherical Bogoliubov–Poisson, and axisymmetric real-time calculations test the predicted energy corrections and vortex matching, and resolve finite-size surface-mode minima whose approach to the continuum surface coefficient is followed to $N\simeq 4\times {10}^6$. For the reference gradient coefficient the vortex constant is $c_1=-0.535$, and the surface Landau scale is $1.417{\omega }_K({\mathrm{\hslash }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^2\mathrm{/}2R_{\mathrm{T}\mathrm{F}}{)}^{1\mathrm{/}6}\propto N^{-1\mathrm{/}9}$; sector energies at $N\simeq 4.6\times {10}^5$, computed after the fit, agree with the extrapolated vortex threshold to ${10}^{-4}$. These results distinguish the accuracy of the Thomas–Fermi bulk from the gradient physics that controls the surface and vortex core, and separate vortex energetics from the open problem of nonlinear vortex entry.