The Faddeev and Schrödinger approaches to Efimov states—a numerical comparison

Abstract

We compare effective hyperradial three-body potentials calculated using the S-wave part of the Faddeev equations to calculations using the full Schrödinger equation. As two-body model potential we test both a short-range potential and a Lennard-Jones potential with van der Waals tail. In the former case we find excellent agreement between the two methods for the lowest adiabatic state, indicating that the Faddeev method can be a useful tool also for numerical computations. For excited states the two methods show important differences, but agree for hyperradii larger than about five times the range of the potential (independent of the value of the scattering length and of the number of bound states). For the van der Waals potential, we focus on how well the Faddeev method reproduces the so-called van der Waals universality. We find that indeed the universality is manifest also using this method, but at a slightly different value of the universal parameter . We further derive an efficient method to solve the integro-differential equation arising in the Faddeev method.