Solving a general three-body atomic problem with Generalized Sturmian Functions

The quantal description of few body systems is of fundamental interest in atomic physics. The complete dynamics is contained in the wave function, the solution of the Schro ̈dinger equation together with appropriate boundary conditions. Unfortunately, the mentioned solutions are exactly known for few particular interaction potentials, and only for the two-body case. For more-than-two particle systems, it has to be appealed to numerical procedures. Between the many existing approximate techniques, an ab-initio methodology is the one which allow the evaluation of the solution to arbitrary precision, if the demanded computational resources are available. The major obstacle in the de- velopment of such a methodology is the proper imposition of the asymptotic behaviour, when stationary scattering states are considered.
In this presentation we will see the characteristics of an ab-initio methodology, able to deal with two important kinds of three body systems: the calculation of the structure of bounded states (atomic and molecular states), and the evaluation of scattering states, like the one in which a two body bounded system is fragmented by the impact of a third particle (e. g. hydrogen ionisation by electron impact), or a three body system which is fragmented by the incidence of light (e. g. double-photoionisation of helium). The quantity of interest of a collisional process is the cross section, which can be extracted from the scattering wave function.
The resolution of the Schr ̈odinger equation for the mentioned problems is performed trough a Sturmian expansion of the wave function and a subsequent evaluation of the expansion’s coefficients by means of the Galerkin method. The Sturmian functions came from a two body eigenvalue (model) problem, which allow an easy imposition of atomic boundary behaviours, such as Kato cusp conditions, or outgoing (incoming) flux condi- tions (adequate these last ones for the evaluation of stationary scattering states). We will show the set of systems which can be studied at this stage of development of the codes.