# Bound states and scattering length

1. What is the relationship between bound states and scattering length?

2. What is the relationship between scattering states and scattering length?

3. When we say, potential is 'like' repulsive for positive scattering length and viceversa, are we talking with respect to scattering states or bound states? (though the answer should be scattering states, but the literature everywhere assumes you to know it on your own.)

All these concepts are found in the theory of BCS-BEC crossover.

By the magic of analyticity there are some relations between binding energies and phase shifts. One example is Levinson's theorem (http://ajp.aapt.org/resource/1/ajpias/v32/i10/p787_s1). Another important example has to do with shallow bound states. If there is a shallow bound state with energy $E=-E_B$, then the s-wave scattering length a satisfies $E_B=1/(ma^2)$. This is explained in most text books on QM (I was perusing Weinberg's book earlier this year, and he has a whole section devoted to shallow bound states.)
1) For shallow bound states, $E_B=1/(ma^2)$. In general, no direct relation except for global'' statements like Levinson's theorem.
3) We call negative $a$ attractive and positive $a$ repulsive because the asymptotic wave functions are pulled in or pushed out, respectively. Also, the simplest mechanism for small negative/positive $a$ is a weak repulsive/attractive potential. If there is a shallow bound state then $a$ is positive. This means that the scattering wave is pushed out (repulsive'') even though the underlying potential is obviously attractive. This means that at low energy one cannot distinguish scattering from weakly repulsive potentials and strongly attractive ones with a shallow bound state.