= Amplitude poles and analytic continuation of transmission probability
{title2=$T(k)=t(k)t(-k)$}
For a real one-dimensional potential, a real-axis <transmission probability> is a modulus square. Its algebraic complex continuation need not have only physical <Bound-state pole>[bound-state poles]. In a square well, $t=e^{-ika}/D(k)$ while the continued <probability> is $1/[D(k)D(-k)]$. Zeros of the second factor can encode a different <boundary condition>. In units $\hbar^2/(2m)=1$, a well of width two and depth $1+\cot^2 1$ has a continued-probability <pole> at $k=i\cot1$ which satisfies neither decaying even nor decaying odd matching. The physical bound-state criterion uses the outgoing <amplitude> on its decaying sheet.
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