= Bound-state thresholds for a square well with one hard wall
{title2=$\mathcal U_j=\hbar^2\pi^2(j+1/2)^2/(2ma^2)$}
For a zero-potential interval of width $a$, an infinite wall at its left end, and constant exterior potential $U_0$ to its right, the $j$th <bound state> branch exists precisely when $U_0>\mathcal U_j$, for $j=0,1,\ldots$. Matching the <wavefunction> and its derivative gives $z\cot z=-\sqrt{z_0^2-z^2}$ with $z_0=a\sqrt{2mU_0}/\hbar$. At equality the new threshold solution is not <normalizable>, so exactly one bound state corresponds to $U_0\in(\mathcal U_0,\mathcal U_1]$.
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