  recently proposed strictly phenomenological static quark-antiquark potential
belonging to the generality $V(r)=-Ar^{-\alpha}+\kappa r^{\beta}+V_{0}$ is
tested with heavy quarkonia in the context of the shifted large N-expansion
method. This nonrelativistic potential model fits the spin-averaged mass
spectra of the $c\bar{c},$ $b\bar{b}$ and $c% \bar{b}$ quarkonia within a few
${\rm MeV}$ and also the five experimentally known leptonic decay widths of the
$c\bar{c}$ and $b% \bar{b}$ vector states. Further, we compute the hyperfine
splittings of the bottomonium spectrum as well as the fine and hyperfine
splittings of the charmonium spectrum. We give predictions for not yet observed
$B_{c}$ splittings. The model is then used to predict the masses of the
remaining quarkonia and the leptonic decay widths of the two pseudoscalar
$c\bar{b%}$ states. Our results are compared with other models to gauge the
reliability of the predictions and point out differences.
