Upper and lower limits on the number of bound states in a central potential

TL;DR

This paper extends and refines bounds on the number of bound states in quantum central potentials, providing new upper and lower limits for various partial waves and potential types, including non-monotonic and positive potentials.

Contribution

It generalizes previous bounds to all partial waves and non-monotonic potentials, introducing new, tighter lower bounds for the number of bound states in quantum systems.

Findings

Derived new lower bounds for the number of bound states in various partial waves.

Extended bounds to non-monotonic and positive potentials.

Provided a neat lower limit for the total number of bound states.

Abstract

In a recent paper new upper and lower limits were given, in the context of the Schr\"{o}dinger or Klein-Gordon equations, for the number $N_{0}$ of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number $N_{\ell}$ of bound states for the $\ell$-th partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat \textit{lower} limit $N_{\ell}\geq \{\{[\sigma /(2\ell+1)+1]/2\}\}$ with $% \sigma =(2/\pi) \underset{0\leq r<\infty}{\max}[r| V(r)| ^{1/2}]$ (valid in the Schr\"{o}dinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following \textit{lower} limit for the total number $N$ of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: $N\geq \{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2$ (here the double braces denote of course the integer part).