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We study in dimension $d\geq2$ low-energy spectral and scattering asymptoticsfor two-body $d$-dimensional Schr\"odinger operators with a radially symmetricpotential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angularmomentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2-1)^2$. Ineach such sector the reduced Schr\"odinger operator has infinitely manynegative eigenvalues accumulating at zero. We show that the resolvent has anon-trivial oscillatory behaviour as the spectral parameter approaches zero incones bounded away from the negative half-axis, and we derive an asymptoticformula for the phase shift.

Two-body threshold spectral analysis, the critical case