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We consider, for $h,E>0$, the semiclassical Schr\"odinger operator$-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and itsradial derivative $\dell_{r}V$ are bounded away from the origin, havelong-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while$\dell_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta\leq4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound isexponential in $h^{-1}$, while the exterior resolvent bound is linear in$h^{-1}$.

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