Two-body threshold spectral analysis, the critical case

We study in dimension $d\ge 2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schr"odinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\gamma >0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma >(l+d\mathrm{/}2-1{)}^2$. In each such sector the reduced Schr"odinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.