 Half-line finite-well bound-state count

For dimensionless depth $\nu =a\sqrt{2m{V}_0}/\hslash$, each negative branch of $z\cot z=-\sqrt{{\nu }^2-z^2}$ gives one bound state. The $n$th state exists exactly when $\nu >(n-1/2)\pi$. At equality its exterior decay constant vanishes and the state is not square integrable. In particular, exactly two bound states means $3\pi /2<\nu \le 5\pi /2$. This strict lower and inclusive upper endpoint convention counts negative-energy eigenstates, excluding a zero-energy threshold solution.