A simple interpolation formula for the spectra of power-law and log potentials

TL;DR

The paper introduces a unified approach to spectra of power-law and logarithmic potentials, deriving a simple interpolation formula that accurately estimates eigenvalues across these potential types.

Contribution

It presents a novel interpolation formula that smoothly connects power-law and logarithmic potential spectra, providing highly accurate eigenvalue estimates.

Findings

The interpolation formula accurately predicts the first thirty eigenvalues with less than 0.04% error.

Spectra of power-law and logarithmic potentials are unified within a single family.

Eigenvalues can be smoothly obtained by taking the limit q --> 0 in the power-law spectrum.

Abstract

Non-relativistic potential models are considered of the pure power V(r)=sgn(q) r^q and logarithmic V(r)=ln(r) types. It is shown that, from the spectral viewpoint, these potentials are actually in a single family. The log spectra can be obtained from the power spectra by the limit q --> 0 taken in a smooth representation P(n,l,q) for the eigenvalues E(n,l,q). A simple approximation formula is developed which yields the first thirty eigenvalues with error < 0.04%.