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

Non-relativistic potential models are considered of the pure power V (r) = sgn(q)r 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 Pnl(q) for the eigenvalues Enl(q). A simple approximation formula is developed which yields the first thirty eigenvalues with error < 0.04%. PACS 12.39.Pn; 12.39.Jh; 03.65.Ge. A simple interpolation formula for the spectra of power-law and log potentials page 2