Generalization of Legendre functions applied to Rosen-Morse scattering states

TL;DR

This paper introduces a generalized form of Legendre functions to analyze Rosen-Morse potential scattering states, providing explicit formulas, asymptotic analysis, and a complete classical solution without path integrals.

Contribution

It proposes a new generalization of Legendre functions, deriving explicit formulas and integral identities for scattering analysis of the Rosen-Morse potential.

Findings

Derived explicit formulas for generalized Legendre functions

Analyzed asymptotic behavior matching scattering state requirements

Provided elementary expressions for reflection and transmission coefficients

Abstract

A generalization of associated Legendre functions is proposed and used to describe the scattering states of the Rosen-Morse potential. The functions are then given explicit formulas in terms of the hypergeometric function, their asymptotic behavior is examined and shown to match the requirements for states in the regions of total and partial reflection. Elementary expressions are given for reflection and transmission coefficients, and an integral identity for the generalized Legendre functions is proven, allowing the calculation of the spectral measure of the induced integral transform for the scattering states. These methods provide a complete classical solution to the potential, without need of path integral techniques.