Two-body threshold spectral analysis, the critical case

We study in dimension d ⩾ 2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrodinger operators with a radially symmetric potential falling off like − γ r − 2 , γ > 0 . We consider angular momentum sectors, labelled by l = 0 , 1 , … , for which γ > ( l + d / 2 − 1 ) 2 . In each such sector the reduced Schrodinger 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.