Integro-differential equation for Bose-Einstein condensates

摘要

We use the assumption that the potential for the A-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for A-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials P_k~α β(x) and of the weight function W(z) which give severe numerical problems for very large A. However, using appropriate limits for A → ∞ we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for A ∈ (10-100) ~(87)Rb atoms interacting via interatomic interactions and confined by an externally applied trapping potential V_(trap)(r). Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.