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 $-h^2\mathrm{\Delta }+V-E$ in dimension two and higher. The potential $V$, and its radial derivative ${\text{dell}}_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta }$ near the origin while ${\text{dell}}_{r}V$ is bounded by $r^{-1-\delta }$, where $0\le \delta \le 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.