Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.