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

TL;DR

This paper derives an analytical formula for the threshold energy of the most weakly bound-state resonance in quantum systems with inverse power-law potentials and a short-range core, enhancing understanding of highly excited bound states.

Contribution

It provides a new analytical expression for the threshold energy of weakly bound states in inverse power-law potentials with a short-range core, which was not previously available.

Findings

Derived a compact analytical formula for threshold energy

Characterized the most weakly bound-state resonance

Applicable to potentials with inverse power-law tails

Abstract

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.