Bound states in propagator

[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?

 

[azntoon]

The reason why is that a bound state only exists when the energy of the system is negative, and this can only occur when the pole of the scattering amplitude is in the upper-half of the complex wave-number plane. This is because the energy of the bound state must be less than zero, which implies that the real part of the wave number must be negative. The upper-half of the complex wave-number plane is defined by having a positive imaginary part, so if the pole is in this region then the real part of the wave number will be negative and thus the energy of the system will be negative, resulting in a bound state.