Lower limit in semiclassical form for the number of bound states in a central potential

TL;DR

This paper establishes a rigorous lower bound for the number of bound states in certain central potentials using a semiclassical estimate, extending to higher angular momentum states and discussing upper bounds for specific potentials.

Contribution

The paper proves that for a class of potentials, the semiclassical estimate provides a rigorous lower limit for the number of bound states, including higher partial waves.

Findings

Semiclassical estimate $N^{(semi)}$ is a lower bound for the number of bound states.

Extension to higher partial waves via effective potential substitution.

An upper limit is also derived for a restricted class of potentials.

Abstract

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}{\pi}\int_0^\infty dr\sqrt{-V(r)\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {{N^{\text{(semi)}}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.