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

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.

References in corpus (1)

• Upper and lower limits for the number of S-wave bound states in an attractive potential

Cited by in corpus (3)

• Necessary and sufficient conditions for existence of bound states in a central potential
• Upper and lower limits on the number of bound states in a central potential
• Lower limit in semiclassical form for the number of bound states in a central potential