Benchmark Kantorovich calculations for three particles on a line

Abstract

A Kantorovich approach is used to solve for the eigenvalue and the scattering properties associated with a multi-dimensional Schrödinger equation. It is developed within the framework of a conventional finite element representation of solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated in the case of an exactly solvable 'benchmark' model of three identical particles on a line, with zero-range attractive pair potentials and below the three-body threshold. In this model all the 'effective' potentials, and 'coupling matrix elements', of the set of resulting close-coupling radial equations, are calculated using analytical formulae. Variational formulations are developed for both the bound-state energy and the elastic scattering problem. The corresponding numerical schemes are devised using a finite element method of high order accuracy.