Algebraic model for quantum scattering: Reformulation, analysis, and numerical strategies

Abstract

The convergence problem for scattering states is studied in detail within the framework of the Algebraic Model, a representation of the Schrödinger equation in an L 2 basis. The dynamical equations of this model are reformulated featuring new "Dynamical Coefficients", which explicitly reveal the potential effects. A general analysis of the Dynamical Coefficients leads to an optimal basis yielding well converging, precise and stable results. A set of strategies for solving the equations for non-optimal bases is formulated based on the asymptotic behaviour of the Dynamical Coefficients. These strategies are shown to provide a dramatically improved convergence of the solutions.

A microscopic three-cluster model with nuclear polarization applied to the resonances of 7Be and the reaction 6Li(p, 3He)4He

“…It was shown that an acceptable precision for light p-shell nuclei can be achieved with 30-50 oscillator functions [10,13,46]. In some model situations this number can be further reduced, even down to 3 or 5 functions, as was shown in [47]. The calculation of matrix elements of different operators between oscillator functions can be done with the technique of the Generalized Coherent States [10,48,49], which leads to recurrence relations for the matrix elements.…”

Algebraic model for scattering in three-s-cluster systems. I. Theoretical background

“…It was conjectured (see for instance [20]) that for very large values of the oscillator quantum number n the expansion coefficients for physically relevant wave-functions behave like…”

“…It dramatically improves the convergence of the results with significantly smaller values for N . We refer to [20] for further details.…”

“…We will now use a representation of the dynamical equations presented in [20]. As we will consider an S-matrix formulation of the problem, the expansion coefficients are rewritten as…”

“…In reference [20] a numerical strategy to account for long tails in the potential in the AM was developed. It dramatically improves the convergence of the results with significantly smaller values for N .…”