G. Gasaneo

Derivatives of any order of the confluent hypergeometric function F11(a,b,z) with respect to the parameter a or b

The derivatives to any order of the confluent hypergeometric (Kummer) function F=F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampe de Feriet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter.

Derivatives of any order of the Gaussian hypergeometric function 2F1(a, b, c; z) with respect to the parameters a, b and c

The derivatives to any order of the Gaussian hypergeometric function 2F1(a, b, c; z) with respect to the parameters a, b and c are expressed in terms of generalizations of multivariable Kampe de Feriet functions. Several properties are presented. In an application to the two-body Coulomb scattering problem, the usefulness of these derivatives is illustrated with the study of the charge dependence of Pollaczek-like polynomials.

Derivatives of any order of the hypergeometric function pFq(a1, ..., ap; b1, ..., bq; z) with respect to the parameters ai and bi

The derivatives of any order of the general hypergeometric function pFq(a1, ..., ap; b1, ..., bq; z) with respect to the parameters ai or bi are expressed, in compact form, in terms of generalizations of multivariable Kampe de Feriet functions. To achieve this, use is made of Babisters solution to non-homogeneous differential equations for pFq(a1, ..., ap; b1, ..., bq; z). An application to Hahn polynomials, which are 3F2 functions, is given as an illustration.

Hyperspherical adiabatic eigenvalues for zero-range potentials

The scattering length a associated with two-body interactions is the relevant parameter for near threshold processes in cold atom-atom collisions. For this reason zero-range potentials are traditionally used to model collective behaviour of dilute collections of bosons. The model is also used to compute three-body recombination rates, where it gives an a4 law. In this paper we examine the applicability of the zero-range model to real physical systems. Hyperspherical adiabatic potentials obtained from the zero-range model are compared with published potentials based on realistic two-body interactions. From these comparisons it is possible to determine the regions where the model applies.

Two-body Coulomb wavefunctions as kernel for alternative integral transformations

In this paper we investigate different representations of an arbitrary function in terms of two-body Coulomb eigenfunctions. We discuss the standard energy basis in spherical and parabolic coordinates with the purpose of remarking explicitly that two additional parameters appear both in the Schrodinger equation and in the wavefunctions: the charge and the angular momentum. We introduce the charge and generalized angular momentum Sturmian function representations, which result when the charge or the angular momentum is used as the eigenvalue in the Coulomb Schrodinger equation, respectively. We present the connection between the generalized angular momentum representation and the Kontorovich–Lebedev transform. Finally, we extend the angular momentum representation to six dimensions, which is suitable for further applications in the three-body Coulomb problem.

A Kontorovich–Lebedev representation for zero-range potential eigensolutions

A novel mathematically simple Kontorovich–Lebedev representation of solutions to the Schrodinger equation for a three-particle problem where two of them interact via a zero-range potential is developed. The asymptotic limits and regularity properties are studied. The connection between the representations for E > 0 and E < 0 is also discussed.

Electron-ion correlation effects in ion-atom single ionization

We study the effect of electron-ion correlation in single ionization processes of atoms by ion impact. We present a distorted wave model where the final state is represented by a correlated function solution of a non-separable three-body continuum Hamiltonian, that includes electron-ion correlation as coupling terms of the wave equation. A comparison of the electronic differential cross sections computed with this model with other theories and experimental data reveals that the influence of the electron-ion correlation is more significant for low energy emitted electrons.

A spectral approach based on generalized Sturmian functions for two- and three-body scattering problems

A methodology based on generalized Sturmian functions is put forward to solve two- and three-body scattering problems. It uses a spectral method which allows for the inclusion of the correct asymptotic behavior when solving the associated driven Schr?dinger equation. For the two-body case, we demonstrate the equivalence between the exterior complex scaling (ECS) and the Sturmian approaches and illustrate the latter by using Hulth?n Sturmian functions. Contrary to the ECS approach, no artificial cut-off of the potential is required in the Surmian approach. For the three-body scattering problem, the theoretical framework is presented in hyperspherical coordinates and a set of hyperspherical generalized Sturmian functions possessing outgoing asymptotic behavior is introduced. The Sturmian procedure is a direct generalization of the method discussed for the two-body problem; thus, the comparison with the ECS method is similar. For both the two- and three-body cases, Sturmian bases are efficient as they possess the correct outgoing behavior, diagonalize part of the potentials involved and are essentially localized in the region where the unsolved interaction is not negligible. Moreover, with the Sturmian basis, the operator (H ? E) is represented by a diagonal matrix whose elements are simply the Sturmian eigenvalues.

Integral representation of one-dimensional three particle scattering for δ function interactions

The Schrodinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrodinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger n...

A special asymptotic limit of a Kampe de Feriet hypergeometric function appearing in nonhomogeneous Coulomb problems

In the investigation of two-body Coulomb Schrodinger equations with some types of nonhomogeneities, the particular solution can be expressed in terms of a two-variable Kampe de Feriet hypergeometric function. The asymptotic limit of the latter—for both variables being large but their ratio being a bound constant—is required in order to extract relevant physical information from the solutions. In this report the mathematical limit is provided. For that purpose, a particular series representation of the hypergeometric function—in terms of products of Kummer and Gauss functions—is first derived.