Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity

TL;DR

This paper derives exact solutions for scattering and bound states in novel hyperbolic potentials with inverse-square singularities using the Tridiagonal Representation Approach, expanding in Jacobi and Wilson polynomials.

Contribution

It introduces a new application of the TRA to hyperbolic potentials with inverse-square singularities, providing explicit series solutions for bound and scattering states.

Findings

Finite series for bound states

Infinite but bounded series for scattering states

Explicit solutions in terms of Jacobi and Wilson polynomials

Abstract

We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schr\"odinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite, but bounded, for the continuum scattering states.