Unitary gas in an isotropic harmonic trap: Symmetry properties and applications

We consider $N$ atoms trapped in an isotropic harmonic potential, with $s$-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, existence of a decoupled bosonic degree of freedom, and relations between the moments of the trapping potential energy and the moments of the total energy.