The quantum scattering amplitude in Rn for potentials decreasing faster than ‖x‖−(n+1)/2

Abstract

Within the framework of the two-body problem in quantum mechanics, we prove an ‘‘analytic limiting absorption principle’’ for the Schrödinger operator H=H0+V, on L2(Rn), where H0=−Δ and V is a real short-range potential, that is, decreasing like ‖x‖−1−ε, ε>0, as ‖x‖→∞. Applying it in the case where V decreases like ‖x‖−(n+1)/2(log‖x‖)−1−ε, ε>0, as ‖x‖→∞, we obtain two classes of results for the non-Born part of the scattering amplitude: finiteness, continuity and high energy properties on the positive real axis, including correction to the Born approximation of the cross section; analyticity and high energy properties in the upper half-plane for the forward amplitude.