Energy spectrum design and potential function engineering

TL;DR

This paper introduces a method to design energy spectra using orthogonal polynomials and numerically derives the associated potential functions, exemplified with the continuous dual Hahn polynomial, advancing quantum mechanics modeling.

Contribution

It presents a novel approach to energy spectrum design via orthogonal polynomials and provides a numerical method to obtain potential functions from the spectrum.

Findings

Explicit energy spectrum formulas for the continuous dual Hahn polynomial

Numerical potential functions corresponding to the chosen spectra

Analytic expressions for bound states and phase shifts

Abstract

Starting with an orthogonal polynomial sequence $\{p_n(s)\}_{n=0}^\infty$ that has a discrete spectrum, we design an energy spectrum formula, $E_k = f (s_k)$, where $|{s_k\}$ is the finite or infinite discrete spectrum of the polynomial. Using a recent approach for doing quantum mechanics based, not on potential functions but, on orthogonal energy polynomials, we give a local numerical realization of the potential function associated with the chosen energy spectrum. In this work, we select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound states energy spectrum, scattering states phase shift, and wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.