In the previous sections we considered situation, when the scattering happens on the
lowest energy level. In this case the -wave scattering length
for any finite
strength potential is smaller than the range of the potential
(see, e.g.
Secs. 1.3.2.3,1.3.3.3) and equals to
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
. In this sense the range of the
pseudopotential is zero
and we have opposite condition
A physical realization of 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.