Pseudopotential in resonant regimes

The zero-range potential approach is extended for the description of situations where two-body scattering is resonant in arbitrary partial waves. The formalism generalizes the Fermi pseudopotential which can be used only for $s$-wave broad resonances. In a given channel, the interaction is described either in terms of a contact condition on the wave function or with a family of pseudopotentials. We show that it is necessary to introduce a regularized scalar product for wave functions obtained in the zero-range potential formalism (except for the Fermi pseudopotential). This metrics shows that the geometry of these Hilbert spaces depends crucially on the interaction.

Figure 1

Schematic representation of a two-body effective radial potential $V_l\left(r\right)=U\left(\vec{r}\right)+{\hslash }^{2}l(l+1)∕2\mu r^2$ in a partial wave $l>0$. $E_r$ is the energy of a quasibound state [4, 38] which leads to a resonant behavior in the two-body scattering for a collisional energy $E\simeq E_r$.