A class of ($\ell$-dependent) potentials with the same number of   ($\ell$-wave) bound states
  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. 
Fabian Brau and Francesco Calogero