= Solution
On the physical sheet the bound-state <amplitude> <Pole>[poles] lie at $k=i\kappa_j$, $\kappa_j>0$, and map to real negative $E_j=-\hbar^2\kappa_j^2/(2m)$. The square-root map has two sheets meeting at the continuum threshold. <Analytic continuation> can also have virtual-state <Pole>[poles] on the negative imaginary $k$ axis and <Resonance pole>[resonance poles] in the <lower half-plane>; these are not normalizable <Bound state>[bound states]. In the <energy> plane the <resonance (chemistry)>[resonances] belong to the nonphysical sheet, rather than additional negative-axis bound levels.
For the algebraically continued \i[<probability>], include the mirror <Pole>[poles] contributed by $D(-k)$ as explained in part d; the two $k$ signs map to the same numerical <energy> but need not impose the same <boundary condition>. A sketch must label the sheet or the <amplitude> being plotted, rather than treating every singularity as a <bound state>.
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