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

We consider, for h, E>0, the semiclassical Schrödinger operator -h^2 + V - E in dimension two and higher. The potential V and its radial derivative ₑV are bounded away from the origin, have long-range decay and V is bounded by r^- near the origin while ₑV is bounded by r^-1-, where 0 < 4 (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.