A SERIES REPRESENTATION OF THE THREE BODY COULOMB GREEN'S FUNCTION FOR POSITIVE ENERGIES

A. L. FRAPICCINI, J. M. RANDAZZO, F. D. COLAVECCHIA AND G. GASANEO

Freiburg, Alemania

Conferencia; XXV ICPEAC, International Conference on Photonic, Electronic and Atomic Collisions; 2007

The radial part of the Coulomb Green's function (CGF) for a two-body system can be easily represented by a sum of properly normalized Sturmian functions at negative energies. Several methods have been developed to extend this representation to positive energies. The main difficulty of these procedures is to find a discrete, convergent basis set that includes the correct Coulomb asymptotic behavior. Therefore, it would be possible to write the CGF as a sum over the Sturmian eigenvalues, in the same way as the negative energies situation. In this report we present a discrete set of Sturmians functions that is regular at the origin and have the correct asymptotic behavior according to the energy domain: exponential decay at negative energies, and outgoing, incoming or standing wave condition at positive energies. These Sturmian functions are used to write expressions for the three--body Coulomb Green's functions xorresponding to the independent electron model.Making use of the discrete set of positive energy, outgoing Sturmian functions, we can construct a radial three-body Sturmian basis set. The Sturmian basis set is obtained by two different ways: expanding the function in a finite, L2 Laguerre-type basis set or by direct discretization of the Hamiltonian. In both cases the correct asympotic behavior is imposed. The Sturmian basis set  is used to expand the radial part of the three-body Coulomb Green's function for noninteracting electrons.The coefficients are obtained with both representations of the Sturmian basis set. The numerical efficiency of the methods is discussed for the different asymptotic conditions studied both for positive and negative energies.