A Hyperspherical Treatment of the N-Fermion problem.

Hyperspherical methods provide for an interesting approach to studying the N body problem. ~We develop this unconventional description for the ground state and collective oscillations of the two-component normal Fermi gas with two-body s and p-wave contact interactions in an isotropic trap. ~The many-body problem can be accurately reduced to a linear, one-dimensional Schr\"{o}dinger equation in a single collective coordinate, the hyperradius (the root mean square radius) R of the N-atom system. ~The calculated properties of the Fermi gas ground state~are shown to be in close agreement with results from the Hartree-Fock (HF) approximation over a wide range of interspecies scattering lengths while the collective breathing mode excitation energy deviates qualitatively from HF predictions. ~The hyperspherical method also suggests that the Fermi gas may collapse for sufficiently large and negative scattering lengths.