Highly excited bound-state resonances of short-range inverse power-law potentials

We study analytically the radial Schr\"odinger equation with long-range attractive potentials whose asymptotic behaviors are dominated by inverse power-law tails of the form $V(r)=-\beta_n r^{-n}$ with $n>2$. In particular, assuming that the effective radial potential is characterized by a short-range infinitely repulsive core of radius $R$, we derive a compact {\it analytical} formula for the threshold energy $E^{\text{max}}_l=E^{\text{max}}_l(n,\beta_n,R)$ which characterizes the most weakly bound-state resonance (the most excited energy level) of the quantum system.