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 -h²Δ + V - E in dimension two and higher. The potential V and its radial derivative ∂ᵣV are bounded away from the origin, have long-range decay and V is bounded by r-δ near the origin while ∂ᵣV is bounded by r-1-δ , where 0≤δ < 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 .