Positive energy Sturmians in a L2 basis for scattering problems

A.L. FRAPICCINI, G. GASANEO, F.D. COLAVECCHIA , AND D. MITNIK

Santos, Brasil

Conferencia; FB18: 18th International Conference on few body problems in Physics; 2006

Sturmian functions [1-2] have been widely used as a basis set in atomic physics calculations, e.g., to expand the Coulomb Green function [3] or to determine atomic energy levels [4-6]. This Sturmian basis set is chosen to satisfy certain boundary conditions, such as regularity at the origin and the correct asymptotic behavior according to the energy domain: exponential decay (bound states) for negative energy and outgoing (or incoming or standing wave) boundary condition for positive energy. However, Sturmian functions are often hard to compute, specially in the positive energy case.The advantage of using Sturmian functions as a basis set is to provide the correct boundary conditions to the solution of scattering equations [7]. In this report, we introduce a method to obtain Sturmian functions by solving the radial part of theSchrödinger equation. In the positive energy case, we seek for the solutions with outgoing wave boundary conditions. The method here proposed consist on expanding the Sturmian function in terms of a finite L2 Laguerre-type basis set. Convergence is achieved as N increases. In order to check our results, two alternative methods have been developed: in the first one the Schrodinger equation, is solved by direct diagonalization of the matrix representing the full Hamiltonian in a radial lattice basis. The new feature of our method, is that this matrix is modified (it becomes a complex matrix), in sucha way that it includes the desired outgoing condition at the boundary. The second approach uses a Numerov shooting technique. We numerically solve the equation in the interior region and obtain the eigenvalues by matching the interior solution withthe incoming or outgoing boundary condition at a distance R greater than the range of the potential V (r).