# Bound states in propagator

1. Oct 13, 2012

### geoduck

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?

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

$$A=\frac{1}{|k|-iB}$$
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?