The Schrodinger equation incorporating the long-range Coulomb potential takes the form of a Fredholm equation whose kernel is singular on its diagonal when represented by a basis bearing a continuum of states, such as in a Fourier-Bessel transform. S
everal methods have been devised to tackle this difficulty, from simply removing the infinite-range of the Coulomb potential with a screening or cut function to using discretizing schemes which take advantage of the integrable character of Coulomb kernel singularities. However, they have never been tested in the context of Berggren bases, which allow many-body nuclear wave functions to be expanded, with halo or resonant properties within a shell model framework. It is thus the object of this paper to test different discretization schemes of the Coulomb potential kernel in the framework of complex-energy nuclear physics. For that, the Berggren basis expansion of proton states pertaining to the sd-shell arising in the A ~ 20 region, being typically resonant, will be effected. Apart from standard frameworks involving a cut function or analytical integration of singularities, a new method will be presented, which replaces diagonal singularities by finite off-diagonal terms. It will be shown that this methodology surpasses in precision the two former techniques.
Is there a connection between the branch point singularity at the particle emission threshold and the appearance of cluster states which reveal the structure of a corresponding reaction channel? Which nuclear states are most impacted by the coupling
to the scattering continuum? What should be the most important steps in developing the theory that will truly unify nuclear structure and nuclear reactions? The common denominator of these questions is the continuum shell-model approach to bound and unbound nuclear states, nuclear decays, and reactions.
The isospin breaking effects due to the Coulomb interaction in weakly-bound nuclei are studied using the Gamow Shell Model, a complex-energy configuration interaction approach which simultaneously takes into account many-body correlations between val
ence nucleons and continuum effects. We investigate the near-threshold behavior of one-nucleon spectroscopic factors and the structure of wave functions along an isomultiplet. Illustrative calculations are carried out for the T=1 isobaric triplet. By using a shell-model Hamiltonian consisting of an isoscalar nuclear interaction and the Coulomb term, we demonstrate that for weakly bound or unbound systems the structure of isobaric analog states varies within the isotriplet and impacts the energy dependence of spectroscopic factors. We discuss the partial dynamical isospin symmetry present in isospin-stretched systems, in spite of the Coulomb interaction that gives rise to large mirror symmetry breaking effects.
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently
immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forreys transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.
