On the Laguerre Representation of Coulomb Functions and the Relation to Orthogonal Polynomials
DEL PUNTA, JESSICA A.; GASANEO, GUSTAVO; ANCARANI, LORENZO U.
ADVANCES IN QUANTUM CHEMISTRY
ELSEVIER ACADEMIC PRESS INC
Año: 2018 vol. 76 p. 79 - 101
We investigate the two-body Coulomb radial problem, providing extensions of known results and establishing a novel connection to orthogonal polynomials. The expansion in Laguerre-type functions of positive energy Coulomb solutions allows one to separate out the radial coordinate from the physical parameters. For the regular Coulomb wave function analytical coefficients are known to be directly connected to Pollaczek polynomials. It turns out that, simultaneously for the attractive and repulsive case, they can also be related to Meixner?Pollaczek polynomials. This allows us to provide a novel interpretation of these coefficients; considering the charge as a variable, we are able to establish orthogonality and completeness properties for these charge functions. We also investigate analytically Laguerre-type expansions of the irregular, incoming and outgoing Coulomb solutions; through a careful limit process we provide the expansion coefficients in closed form.