Semiclassical Resolvent Estimates for Schrödinger Operators with Coulomb Singularities

Abstract

Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h −1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.