Extending the class of solvable potentials: III. The hyperbolic single wave

TL;DR

This paper introduces a new solvable hyperbolic single wave potential by expanding solutions in a special basis, using a novel parameter spectrum approach to determine eigen-energies, and expressing wavefunctions with special polynomials.

Contribution

It presents a new solvable potential and a unique method to find eigenvalues via the parameter spectrum, expanding the class of exactly solvable quantum models.

Findings

New hyperbolic single wave potential identified

Eigen-energies obtained through parameter spectrum inversion

Wavefunctions expressed with Gegenbauer and dipole polynomials

Abstract

A new solvable hyperbolic single wave potential is found by expanding the regular solution of the 1D Schr\"odinger equation in terms of square integrable basis. The main characteristic of the basis is in supporting an infinite tridiagonal matrix representation of the wave operator. However, the eigen-energies associated with this potential cannot be obtained using traditional procedures. Hence, a new approach (the "potential parameter" approach) has been adopted for this eigenvalue problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the "parameter spectrum"). Subsequently, the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum. The bound states wavefunction is written as a convergent series involving products of the ultra-spherical Gegenbauer polynomial in space and a new polynomial in energy, which is a special case of the "dipole polynomial" of the second kind.