Why must it be true that a system that has a bound state must have its scattering amplitude have a pole in the upper half of the complex wave-number plane?(adsbygoogle = window.adsbygoogle || []).push({});

For example, if the scattering amplitude as a function of the initial wave number magnitude |k| is:

[tex]A=\frac{1}{|k|-iB}[/tex]

with B>0, then there is a pole at k=iB which implies the energy is:

E=k^2/2m=-B^2/2m

that is, a negative energy or bound state.

But if B is negative, then the pole is at k=-i|B| and the energy would be the same using the same formula, but this doesn't represent a bound state because for some reason the pole must be in the upper-half of the complex plane. Why is this true?

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# Bound states in propagator

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