Energy-dependent effective interactions for dilute many-body systems

TL;DR

This paper derives an energy-dependent effective two-body interaction for many-body systems, showing its proportionality to phase shifts, and validates the approach through numerical and analytical methods including applications to Rydberg atoms.

Contribution

It introduces a generalized effective interaction proportional to phase shifts, incorporating energy dependence and effective range corrections for mean-field calculations.

Findings

Effective interaction proportional to phase shift.

Numerical agreement with analytical expressions for energy levels.

Generalized Gross--Pitaevskii equation with effective range corrections.

Abstract

We address the issue of determining an effective two-body interaction for mean-field calculations of 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.