Extending the class of solvable potentials: II. Screened Coulomb potential with a barrier

This is the second article in a series where we succeed in enlarging the class of solvable problems in one and three dimensions. We do that by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. Consequently, the wave equation becomes equivalent to a three-term recursion relation for the expansion coefficients of the wavefunction in the basis. Finding solutions of the recursion relation is equivalent to solving the original problem. This method gives a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. However, the tridiagonal requirement allows only very few and special potentials to be added to the solvability class. In the present work, we obtain S-wave solutions for a three-parameter 1/r singular but short-range potential with a non-orbital barrier and study its energy spectrum. We argue that it could be used as a more appropriate model for the screened Coulomb interaction of an electron with extended molecules. We give also its resonance structure for non-zero angular momentum. Additionally, we plot the phase shift for an electron scattering off a molecule modeled by a set of values of the potential parameters.

💡 Research Summary

The paper presents a novel analytical framework for solving a class of quantum‑mechanical scattering and bound‑state problems that go beyond the traditional set of exactly solvable potentials. The authors abandon the usual requirement that the Hamiltonian be represented by a diagonal matrix in a chosen basis. Instead, they construct a complete, square‑integrable orthogonal basis in which the wave operator assumes a tridiagonal (three‑term) matrix form. In such a basis the expansion coefficients of the wavefunction satisfy a three‑term recursion relation. Solving this recursion is mathematically equivalent to solving the original Schrödinger equation, and the recursion can be tackled either analytically (when the coefficients match known orthogonal‑polynomial recursions) or numerically (by imposing normalisation and boundary conditions).

The specific potential investigated is a three‑parameter, short‑range, screened Coulomb interaction with an additional non‑orbital barrier: \