L2 charge Sturmian functions for positive and negative energies

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

Rosario, Argentina

Conferencia; XXIV ICPEAC, International Conference on Photonic, Electronic and Atomic Collisions; 2005

    The two-body radial Coulomb Green function (RCGF) has been widely used in multiphoton calculations[1]. Written in terms of charge Sturmians functions[1], it is diagonal, and can be represented by a series whose discrete index n runs over the charge eigenvalue spectrum. This series representation of the RCGF is of limited use because it diverges when the energy approaches tothe positive real axis. Manakov and co-workers[2] overcome this divergence using a re-expansion of the Sturmians in terms of Laguerre type functions (LF). This suggest that a set of L2 chargeSturmians functions can be introduced, and used to represent the RCGF for arbitrary energy. Two different set of discrete charge Coulomb Sturmian functions are presented and discussed in this report. The representation of the RCGF in terms of these two basis are also analyzed.  The first charge discretization results whenthe solution of the two-body radial is written as an expansion in a Laguerre-type basis set. The spectrum is discrete for both negative and positive energies, and converges to the complete Coulomb spectrum when N → ∞.  Another L2 Sturmian basis set may be obtained by solving the two-body radial Coulomb problem in a box of radius R0. The set of charges for any energy is obtained by forcing the solution to be zero at the boundary r = R0. For negative energy the set of eigencharges is finite and discrete, while an infinite discrete set results for positive energies. The number of states for negative energies tends to infinity whenR0 → ∞. For positive energies the density of levels tends infinity when R0 → ∞.Finally, the RCGF expressed in terms of the  L2 Sturmian functions, which is valid for all energies. We compare the eigencharge spectrum for both sets of Sturmians and study the connection between our RCGF representation and the one given by Manakov.