In this paper we continue our development of a dimensional perturbation theory (DPT) treatment of N identical particles under quantum confinement. DPT is a beyond-mean-field method which is applicable to both weakly and strongly-interacting systems and can be used to connect both limits. In a previous paper we developed the formalism for low-order energies and excitation frequencies. This formalism has been applied to atoms, Bose-Einstein condensates and quantum dots. One major advantage of the method is that N appears as a parameter in the analytical expressions for the energy and so results for N up to a few thousand are easy to obtain. Other properties however, are also of interest, for example the density profile in the case of a BEC,and larger N results are desirable as well. The latter case requires us to go to higher orders in DPT. These calculations require as input zeroth-order wave functions and this paper, along with a subsequent paper, addresses this issue.
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers
N for both small
s-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because
N remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for
N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a
75 shift from the mean-field breathing mode frequency.