Feshbach resonances notwithstanding, most experimentally realized atomic-vapor condensates are dilute and are best described by the mean-field Gross-Pitaevskii equation. For this reason, we use dimensional scaling methods to obtain an analytical approximation to the GP equation that is more accurate and flexible than the commonly used ground-state Thomas-Fermi approximation. We also demonstrate the power of dimensional perturbation theory by providing a full solution of a model BEC Hamiltonian and a two-electron quantum dot Hamiltonian. A feature shared by these examples is the high degree of accuracy provided by the lowest orders of the perturbation theory. In our approach to the full many-body BEC Hamiltonian, we use the lowest orders of many-body dimensional perturbation theory to obtain semi-analytical ground-state energies and collective excitation frequencies. Our many-body calculations for BEC's compare well with the Gross-Pitaevskii results in the weakly-interacting regime, as they should, and are much improved over mean-field theory predictions in the strongly-interacting regime.
Using the perturbation theory,we solve the ground state of Bose-Einstein condensate in composite potential trap,which is composed of one-dimensional linear harmonic oscillator potential and optical lattice potential.We treat the kinetic energy of particles in composite potential trap as a perturbation and analyse the effect of kinetic term as well as composite potential trap on probability density distribution of particles in ground state of condensate.In the end,we infer the main field of application of perturbation theory.