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.

arXiv comment21 pages, 5 figures, 1 table

References in corpus (14)

• Supersymmetry and Quantum Mechanics
• Renormalization of the Inverse Square Potential
• Solution of the nonrelativistic wave equation in the tridiagonal representation approach
• Extending the class of solvable potentials: III. The hyperbolic single wave
• Quantum mechanics without potential function
• Extending the class of solvable potentials: II. Screened Coulomb potential with a barrier
• Representation reduction and solution space contraction in quasi-exactly solvable systems
• Tridiagonal Representation Approach in Quantum Mechanics
• Open problem in orthogonal polynomials
• Orthogonal polynomials derived from the tridiagonal representation approach
• Infinite potential well with a sinusoidal bottom
• Sine function with a cosine attitude
• Solution of an Open Problem about Two Families of Orthogonal Polynomials
• Renormalization of the strongly attractive inverse square potential: Taming the singularity

Cited by in corpus (4)

• Deformed Morse-like potential
• Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity
• Comment on: "Bound states and the potential parameter spectrum". J. Math. Phys. \textbf{67}, 062103 (2020)
• Electrostatic multipole contributions to the binding energy of electrons