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

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.