On the decrease of the number of bound states with the increase of the angular momentum

TL;DR

This paper proves that for certain central potentials, increasing angular momentum reduces the number of bound states, and uses this to derive tighter bounds on the total number of bound states.

Contribution

It introduces a novel proof using supersymmetric quantum mechanics to show the decrease in bound states with increasing angular momentum for specific potentials.

Findings

Increasing angular momentum decreases the number of bound states by at least one.

Provides an improved upper bound on the total number of bound states for the class of potentials.

Validates the theoretical result with mathematical rigor.

Abstract

For the class of central potentials possessing a finite number of bound states and for which the second derivative of $r V(r)$ is negative, we prove, using the supersymmetric quantum mechanics formalism, that an increase of the angular momentum $\ell$ by one unit yields a decrease of the number of bound states of at least one unit: $N_{\ell+1}\le N_{\ell}-1$. This property is used to obtain, for this class of potential, an upper limit on the total number of bound states which significantly improves previously known results.