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 Schrödinger 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 Schrödinger 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. © 2010 Elsevier Inc. All rights reserved.