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A uniform asymptotic approximation of the 3D scattering wavefunction for a central potential: a new Ansatz

A uniform asymptotic approximation of the 3D scattering wavefunction for a central potential is obtained by using a new Ansatz on the form of a solution. This Ansatz employs the regular reduced radial wavefunction of zero orbital angular momentum, generated by the problem itself, rather than a standard special function. The uniform asymptotic approximation of the 3D scattering wavefunction allows finding of the scattering amplitude without using the partial wave expansion. The method is applied to the scattering by a pure Coulomb potential. In contrast to the uniform asymptotic approximation of the 3D Coulomb wavefunction, based on the Airy function that breaks down at the caustic ξ = 0, the uniform asymptotic approximation obtained in this paper is valid over the whole range of the variables. It is shown that the uniform asymptotic approximation of the 3D Coulomb scattering wavefunction obtained through the new Ansatz is the same expression as that of the Gordon solution written in the form obtained by the present authors of a previous paper.