Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin

We consider, for h, E > 0, the semiclassical Schrodinger operator -h(2)Delta + V - E in dimension two and higher. The potential V, and its radial derivative partial derivative V-r are bounded away from the origin, have long-range decay and V is bounded by r(-delta) near the origin while partial derivative V-r is bounded by r(-1-delta), where 0 <= delta < 4(root 2 - 1). In this setting, we show that the resolvent bound is exponential in h(-1), while the exterior resolvent bound is linear in h(-1).