 Amplitude poles and analytic continuation of transmission probability

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 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 ${\hslash }^2/(2m)=1$, a well of width two and depth $1+{\cot }^{2}1$ has a continued-probability pole at $k=i\cot 1$ which satisfies neither decaying even nor decaying odd matching. The physical bound-state criterion uses the outgoing amplitude on its decaying sheet.