= Half-line finite-well bound-state count
{title2=$\nu>(n-\tfrac12)\pi$}
For dimensionless depth $\nu=a\sqrt{2mV_0}/\hbar$, 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 $3\pi/2<\nu\le5\pi/2$. This strict lower and inclusive upper endpoint convention counts negative-energy eigenstates, excluding a zero-energy threshold solution.
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