Integral representation for scattering phase shifts via the phase-amplitude approach

TL;DR

This paper introduces a new integral method for calculating scattering phase shifts using a modified phase-amplitude approach, improving numerical efficiency and applicability to various potentials including Coulomb.

Contribution

It develops a linear envelope equation replacing Milne's nonlinear equation, enabling direct computation of phase shifts from the phase function for any scattering potential.

Findings

Integral representation is highly suitable for numerical implementation.

Method applies to all scattering potentials, including Coulomb.

Provides a general framework for phase shift calculation.

Abstract

A novel integral representation for scattering phase shifts is obtained based on a modified version of Milne's phase-amplitude approach [W.E. Milne, Phys. Rev. $\mathbf{35}$, 863 (1930)]. We replace Milne's nonlinear differential equation for the amplitude function $y$ with an equivalent linear equation for the envelope $\rho=y^2$, which renders the integral representations highly amenable to numerical implementations. The phase shift is obtained directly from Milne's phase function, which in turn is expressed in terms of the envelope function. The integral representation presented in this work is fully general and it can be used for any type of scattering potential, including the Coulomb potential.