J−matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering

Abstract

The J−matrix method introduced previously for s−wave scattering is extended to treat the lth partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L2), Laguerre (Slater), and oscillator (Gaussian) basis sets. The determination of the expansion coefficients of the continuum eigenfunctions in terms of the L2 basis set is shown to be equivalent to the solution of a linear second order differential equation with appropriate boundary conditions, and complete solutions are presented. Physical scattering problems are approximated by a well−defined model which is then solved exactly. In this manner, the generalization presented here treats the scattering of particles by neutral and charged systems. The appropriate formalism for treating many channel problems where target states of differing angular momentum are coupled is spelled out in detail. The method involves the evaluation of only L2 matrix elements and finite matrix operations, yielding elastic and inelastic scattering information over a continuous range of energies.

One- and two-dimensional Coulomb Green's function matrices in parabolic Sturmians basis

“…The second solution q n of the recursion (16) can be obtained from the condition that q n satisfies the same differential equation as the p n [8]. In other words, if p n ∼ 2 F 1 (a, b; c; z),…”

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

“…Indeed, for very large n, the dynamical equations are reduced to a simple (in an oscillator basis a three-term recurrence) form containing the kinetic energy operator solely. The equations can therefore be solved analytically for very large n [2,4,5]. The above mentioned reduction of the equations only involves the supposition that the potential energy matrix elements vanish for very large n, which has been shown to be an acceptable approximation for relatively short-range interactions.…”

“…"irregular" asymptotic coefficients, the explicit form of which can be found in [2,5]. The equations for the asymptotic coefficients in Fourier space are then:…”

“…The analysis of the AM equations presented in this work will be shown to lead to (1) an optimal choice for the basis, given the potential, yielding well-converging and stable solutions of the AM system, or, if the optimal basis cannot be used, to (2) algorithmic procedures for solving the AM system in an acceptable and controlled approximation. These algorithmic procedures will depend on the specific choice made for the basis (i.e.…”

Description of resonant states in the shell model

“…The method is based on the J-matrix formalism in scattering theory [9]. The J-matrix approach utilizes a diagonalization of the Hamiltonian in one of two bases: the so-called Laguerre basis that is of a particular interest for atomic physics applications and the oscillator basis that is appropriate for nuclear physics.…”