Universality in $s$-wave and higher partial wave Feshbach resonances: an illustration with a single atom near two scattering centers

It is well-known that cold atoms near $s$-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial wave Feshbach resonances also have remarkable universal properties. We will illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ge1$, we find $2L+1$ shallow bound states whose energies behave like $1/R^{2L+1}$ when the distance $R$ between the two centers is large. This sheds additional light on the fundamental question whether Efimov effect exists for higher partial wave resonances. The effects of nonresonant partial-wave channels and the shape parameters in the effective range expansions enter as correction terms. Near $p$-wave and higher partial wave resonances, the energies can be described by a simple universal formula in terms of a parameter called ``proximity parameter.'' We will also discuss modifications of the low energy physics due to the long range Van der Waals potential.