Resonance scattering

In the previous sections we considered situation, when the scattering happens on the lowest energy level. In this case the $s$-wave scattering length $a$ for any finite strength potential is smaller than the range of the potential $R$ (see, e.g. Secs. 1.3.2.3,1.3.3.3) and equals to $R$ in the case of the infinite strength potential (Secs. 1.3.2.2,1.3.3.4). The pseudopotential description (Secs. 1.3.3.2,1.3.4.1) falls into a different class of problem used at a small density, where the exact type of the potential is not important and it is substituted by the boundary condition at $r=0$. In this sense the range of the pseudopotential is zero $R=0$ and we have opposite condition

$$\vert a\vert\gg R$$ (1.88)

A physical realization of $3D$ scattering satisfying the condition (1.88) can be achieved in the case of a resonant scattering. In this Section we will describe scattering on the first exited state of attractive potentials supporting a bound state in the case when the position of the excited state is close to zero-energy continuum level.