Bound states of a quartic and sextic inverse-powerlaw potential for all angular momenta

TL;DR

This paper solves the radial Schrödinger equation for a combined quartic and sextic inverse-power-law potential across all angular momenta using a tridiagonal representation approach, revealing finite bound states determined by potential parameters.

Contribution

It introduces a method to find bound states for complex inverse-power-law potentials with mixed singularities for all angular momenta.

Findings

Finite number of bound states depending on potential amplitudes

Solution expressed as a finite series of Bessel polynomials

Applicable to all angular momenta

Abstract

We use the tridiagonal representation approach to solve the radial Schr\"odinger equation for an inverse power-law potential of a combined quartic and sextic degrees and for all angular momenta. The amplitude of the quartic singularity is larger than that of the sextic but the signs are negative and positive, respectively. It turns out that the system has a finite number of bound states, which is determined by the larger ratio of the two singularity amplitudes. The solution is written as a finite series of square integrable functions written in terms of the Bessel polynomial.