Recovering the two-body potential from a given three-body wave function

Abstract

A simple way of deducing the two-body potential from a given two- or three-body wave function is suggested. This method makes it possible to numerically obtain an unknown potential acting between the particles A and B when we know the potentials of their interaction with a third particle C and know the characteristics of the three-body bound state (ABC). Using the examples of the systems (nnp) and (ΛΛα), we show that even very simple three-body wave functions constructed on the basis of the general reasoning and the knowledge of the binding energies and sizes of these systems, allow us to deduce reasonable and realistic nn and ΛΛ potentials. Within this approach, any artificially constructed wave function automatically becomes an exact solution of the corresponding Schrödinger equation with the AB-potential that the method produces. This fact suggests yet another possible application of this method when the AB-potential is known. In such a case we can find a bound state solution of the Schrödinger equation by looking for such values of the free parameters in an artificially constructed wave function that minimize the difference between the deduced and the exact AB-potentials.