Two-body threshold spectral analysis, the critical case

摘要

We study in dimension d≤2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schr?dinger operators with a radially symmetric potential falling off like -γr~(-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 Schr?dinger 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.