A class of ($\ell$-dependent) potentials with the same number of ($\ell$-wave) bound states

TL;DR

This paper introduces a new class of central potentials in nonrelativistic quantum mechanics that uniquely determine the number of bound states for each angular momentum, with explicit formulas and extended variants.

Contribution

It presents a novel class of potentials with a fixed number of bound states independent of angular momentum, including explicit formulas and extensions with specific asymptotic behaviors.

Findings

The potentials have a number of bound states given by a simple formula independent of .

Explicit potential form and parameters are provided for controlling bound state counts.

Extended potentials exhibit  behavior at both origin and infinity.

Abstract

We introduce and investigate the class of central potentials $$V_{\text{CIC}}(g^{2},\mu^{2},\ell,R;r)=-\frac{g^{2}}{R^{2}} (\frac{r}{R})^{4\ell} {[ 1+(\frac{1}{2\ell+1}) (\frac{r}{R})^{2\ell+1}]^{2}-1+\mu^{2}}^{-2}$$, which possess, in the context of nonrelativistic quantum mechanics, a number of $\ell$-wave bound states given by the ($\ell$-independent !) formula $$N_{\ell}^{\text{(CIC)}}(g^{2},\mu ^{2}) ={{\frac{1}{\pi}\sqrt{g^{2}+\mu^{2}-1} (\sqrt{\mu^{2}-1})^{-1} \arctan(\sqrt{\mu^{2}-1})}}$$. Here $g$ and $\mu$ are two arbitrary real parameters, $\ell$ is the angular momentum quantum number, and the double braces denote of course the integer part. An extension of this class features potentials that possess the same number of $\ell$-wave bound states and behave as $(a/r)^{2}$ both at the origin ($r\to 0^{+}$) and at infinity ($r\to \infty$), where $a$ is an additional free parameter.