P-matrix and J-matrix approaches. Coulomb asymptotics in the harmonic oscillator representation of scattering theory

TL;DR

This paper explores the relationship between R- and P-matrix methods and the harmonic oscillator representation in quantum scattering, introducing a discrete P-matrix analogue and applying it to Coulomb asymptotics in nuclear calculations.

Contribution

It develops a discrete P-matrix analogue within the harmonic oscillator framework and demonstrates its application to Coulomb asymptotics in scattering theory.

Findings

Discrete P-matrix is equivalent to the usual P-matrix in the quasiclassical limit.

The approach enables calculation of resonant states and phase shifts in oscillator-based models.

Method effectively incorporates Coulomb asymptotics in nuclear scattering calculations.

Abstract

The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.