In quantum mechanical scattering, there can be bound states and resonances. Both phenomena are connected through analytic continuation of the scattering energy or momentum. This animation shows the connection by tracing poles as the depth of a spherical well gets deeper and deeper.
The above movie exhibits many phenomena of scattering. In the following, a screen shot illustrates some key moments in the animation:
Key moments for the pole trajectories and their transition from resonances to bound states
The left column shows the S-wave T-matrix, |t0|, in the complex-momentum plane (arb. units), the right column shows the phase shift. (a) For a shallow potential, there is no bound state, but only virtual state 1 and resonances 2 and 3. In (b), infinite scattering length is reached which motivates a discussion of universality. In (c), pole 1 became a deeply bound state. Pole 2 and its mirror pole 2’ have met on the imaginary k-axis and then separated again as virtual states 2 and 2', with 2 on its way to become a bound state and 2' a deeper-bound virtual state. Such intriguing S-wave pole trajectories
have only been discovered ten years ago. Levinson’s theorem is also nicely illustrated within this animation.