Energy spectrum design and potential function engineering

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 to 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. We select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound-state energy spectrum, the phase shift of scattering states, and the wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.