Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach

The four exactly-solvable models related to non-sinusoidal coordinates, namely, the Coulomb, Eckart, Rosen-Morse type I and II models are normally being treated separately, despite the similarity of the functional forms of the potentials, their eigenvalues and eigenfunctions. Based on an extension of the prepotential approach to exactly and quasi-exactly solvable models proposed previously, we show how these models can be derived and solved in a simple and unified way.

💡 Research Summary

The paper presents a unified and streamlined derivation of four classic exactly‑solvable quantum‑mechanical models—Coulomb, Eckart, and the two Rosen‑Morse (type I and II) potentials—by extending the prepotential method previously developed for exactly and quasi‑exactly solvable systems. The authors begin by recalling the standard prepotential framework, in which the Schrödinger equation is expressed in terms of a new coordinate z(x) and a prepotential W(z). By choosing z(x) such that its derivative squared, z′(x)², takes one of the four elementary forms 1, z, 1−z, or z(1−z), the method naturally accommodates the four non‑sinusoidal coordinate systems associated with the four potentials.

In this setting the effective potential is given by V(x)=W′(z)²+W″(z). The authors propose a universal ansatz for the prepotential, W(z)=A z + B ln z + C, where A, B, C are model‑dependent constants. Substituting the appropriate z′(x)² into the expression for V(x) reproduces the familiar functional forms: for z′(x)²=1 one obtains the Coulomb potential (−α/x + ℓ(ℓ+1)/x²); for z′(x)²=z one recovers the Eckart potential; for z′(x)²=1−z and z′(x)²=z(1−z) the Rosen‑Morse I and II potentials emerge, respectively.

The energy spectrum follows directly from the algebraic structure of the prepotential. The eigenvalues are simple quadratic functions of the quantum number n and the parameter A, e.g. Eₙ=−(A−n)² (up to additive constants), exactly matching the known results for each model. The corresponding eigenfunctions factorize as ψₙ(x)=φ₀(x) Pₙ(z), where the ground‑state weight φ₀(x)=exp