Generalization of Legendre functions applied to Rosen-Morse scattering states

TL;DR

This work introduces generalized Legendre functions $D^{\mu,\eta}_\nu(x)$ that solve the Rosen-Morse Schrödinger equation and express them via $F(-\nu-\eta,\nu+1-\eta;1-\mu-\eta;\frac{1-x}{2})$ with a prefactor, connecting to standard Legendre functions when $\eta=0$. Through asymptotic analysis, the authors map the parameters to energy and identify two scattering regimes: below the barrier with total reflection and above the barrier with partial transmission, deriving closed-form reflection and transmission coefficients that satisfy $R+T=1$. They prove an integral identity for these generalized functions, enabling a delta-normalized continuum spectral measure $\int \bar{D}^{ip*}(\tanh y) \bar{D}^{ik}(\tanh x)/w_{\alpha,\beta}(k) \, dk = \delta(x-y)$ and a corresponding resolution of the identity, recovering known Legendre results as $\beta\to0$. Overall, the approach provides a complete, purely classical solution to Rosen-Morse scattering and yields simpler spectral-transport expressions than path-integral methods, with potential broader applicability to other analytically solvable potentials.

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.

Figures (2)

• Figure 1: Rosen-Morse scattering state $D^{\mu,\eta}_\nu(\tanh x)$ below the energy barrier. The solution is a distorted wave close to the well, decays into the classically forbidden region as $x\to+\infty$ and shows a wave pattern typical of a wave interfering with its reflection as $x\to-\infty$.
• Figure 2: Rosen-Morse scattering state $D^{\mu,\eta}_\nu(\tanh x)$ above the energy barrier. The solution is a distorted wave close to the well. The frequency at $x>0$ is lower than that at $x<0$. A large portion of the wave is transmitted, so the amplitude increases at $x>0$ to conserve probability current at lower frequency.