The Hahn Quantum System

TL;DR

This paper introduces a new quantum system based on Hahn and continuous Hahn polynomials, linking orthogonal polynomial theory with quantum mechanics to describe scattering states and bound states.

Contribution

It develops a four-parameter quantum system using Hahn polynomials, connecting polynomial asymptotics with physical scattering and bound state phenomena.

Findings

Scattering states expressed via continuous Hahn polynomials

Bound states associated with Hahn polynomials

Asymptotic analysis yields scattering amplitude and phase shift

Abstract

Using a formulation of quantum mechanics based on the theory of orthogonal polynomials, we introduce a four-parameter system associated with the Hahn and continuous Hahn polynomials. The continuum energy scattering states are written in terms of the continuous Hahn polynomial whose asymptotics give the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Hahn polynomial.