Bound states and the potential parameter spectrum

TL;DR

This paper introduces a method to determine the full bound state spectrum and wavefunctions of a wave equation using the potential parameter spectrum, even when the equation is only exactly solvable at a single energy.

Contribution

It develops a novel approach employing orthogonal polynomials and the potential parameter spectrum to find all bound states from limited solvability data.

Findings

The potential parameter spectrum can be finite or infinite.

Bound state energies are obtained by inverting the relation with the PPS.

The method applies to wave equations solvable at a single energy.

Abstract

In this article, we answer the following question: If the wave equation possesses bound states but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite discrete set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.