Ground-state energy of finite Fermi systems

We study the ground-state shell correction energy of a fermionic gas in a mean-field approximation. Considering the particular case of three-dimensional harmonic trapping potentials, we show the rich variety of different behaviors (erratic, regular, supershells) that appear when the number-theoretic properties of the frequency ratios are varied. For self-bound systems, where the shape of the trapping potential is determined by energy minimization, we obtain accurate analytic formulas for the deformation and the shell correction energy as a function of the particle number N. Special attention is devoted to the average of the shell correction energy. We explain why in self-bound systems it is a decreasing (and negative) function of N.