Smooth spectral transition from Coulomb to oscillator

TL;DR

This paper demonstrates a smooth spectral transition between Coulomb and oscillator potentials using envelope representations, providing a unified framework and accurate eigenvalue approximations for various potential models.

Contribution

It introduces a unified approach to connect Coulomb and oscillator spectra through a smooth limit, with a simple approximation formula for eigenvalues and extensions to complex potentials.

Findings

Log spectra derived from power spectra via smooth limit q --> 0

Approximation formula achieves <0.04% error for eigenvalues

Extensions to combined potentials and few-body problems discussed

Abstract

Non-relativistic potential models are considered of the pure power V(r) = sgn(q) r^q and logarithmic V(r) = ln(r) types. Envelope representations and kinetic potentials are employed to show that 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\ell}(q) for the eigenvalues E_{n\ell}(q). A simple approximation formula is developed which yields the first thirty eigenvalues with error < 0.04%. Extensions to potentials with linear combinations of terms such as -a/r + br and applications to spatially-symmetric few-body problems are discussed.