Energy-dependent effective interactions for dilute many-body systems

We address the issue of determining an effective two-body interaction for mean-field calculations of the energies of many-body systems. We show that the effective interaction is proportional to the phase shift and demonstrate this result in the quasiclassical approximation when there is a trapping potential in addition to the short-range interaction between a pair of particles. We calculate numerically energy levels for the case of an interaction with a short-range square well and a harmonic trapping potential and show that the numerical results agree well with the analytical expression. We derive a generalized Gross-Pitaevskii equation which includes effective range corrections and discuss the form of the electron-atom effective interaction to be used in calculations of Rydberg atoms and molecules.