Bound states and the potential parameter spectrum

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.

Deformed Morse-like potential

“…The potential parameter spectrum (PPS) is a non traditional numerical eigenvalue method that originated in the context of the TRA whenever the basis set is energy dependent 11 . This happens, for example, when the TRA constraints dictate that one or more of the basis parameters is/are energy dependent.…”

“…In section III we present the theoretical derivation of the solvable potential (1) using the tridiagonal representation approach. In section IV we compute the energy spectrum associated with this potential using the potential parameter spectrum (PPS), 11 which is a highly non traditional eigenvalue problem. In section IV we confirm the validity of our generated energy spectrum using a direct numerical diagonalization technique in a suitable L 2 basis, 12 and the asymptotic iteration method.…”

Comment on: "Bound states and the potential parameter spectrum". J. Math. Phys. \textbf{67}, 062103 (2020)

“…Summarizing: We have shown that the application of the TRA PPS to the Kratzer-Fues potential [1] has a serious flaw with regard to the values of the radial quantum number k. The resulting expression for the spectrum does not appear to be correct. In addition to it, the well known Frobenius method [2], which resembles the TRA PPS and is taught in undergraduate courses on quantum mechanics and quantum chemistry [6], is far simpler, clearer and more elegant.…”