Non-numerical determination of the number of bound states in some screened Coulomb potentials

We construct potentials ${\mathit{V}}^{\mathrm{\{}\mathit{s}\mathit{c}\mathit{r}\mathit{K}\mathrm{\}}}$(r) which support a finite number scrK of bound states in such a way that the highest normalizable excitation lies precisely at the threshold energy E=0. We expect that many realistic interactions ${\mathit{V}}_{\mathrm{}\left(\mathrm{phys}\right)\mathrm{}}$(r) may be bracketed between these forces due to their flexibility and plausible screened Coulombic form. Thus, we propose an estimate of the number ${\mathit{N}}_{\mathrm{BS}}$ of bound states in an arbitrary potential ${\mathit{V}}_{\mathrm{}\left(\mathrm{phys}\right)\mathrm{}}$(r) via requirements ${\mathit{V}}^{\mathrm{\{}{\mathcal{K}}_{\mathit{L}}\mathrm{\}}}$${\mathit{V}}_{\mathrm{}\left(\mathrm{phys}\right)\mathrm{}}$≥${\mathit{V}}^{\mathrm{\{}{\mathcal{K}}_{\mathit{U}}\mathrm{\}}}$ and subsequent inequalities ${\mathcal{K}}_{\mathit{L}}$≳${\mathit{N}}_{\mathrm{BS}}$≥${\mathcal{K}}_{\mathit{U}}$. In particular, we get a strict prediction ${\mathit{N}}_{\mathrm{BS}}$=${\mathcal{K}}_{\mathit{U}}$ whenever the bracketing proves tight enough, with ${\mathcal{K}}_{\mathit{U}}$≡${\mathcal{K}}_{\mathit{L}}$-1.