Improved resolvent bounds for radial potentials. II

We prove semiclassical resolvent estimates for the Schrödinger operator in $${\mathbb {R}}^d$$ , $$d\ge 3$$ with real-valued radial potentials $$V\in L^\infty ({\mathbb {R}}^d)$$ . show that if $$V(x)={\mathcal O}\left( \langle x\rangle ^{-\delta }\right) $$ $$\delta >4$$ then bound is of form $$\exp \left( Ch^{-\frac{\delta }{\delta -1}}\left( \log (h^{ -1})\right) ^{\frac{1}{\delta -1}}\right) some constant $$C>0$$ If e^{-{\widetilde{C}}\langle ^{\alpha }}\right) $${\widetilde{C}},\alpha >0$$ we get better bounds Ch^{-1}\left( (h^{-1})\right) ^{\frac{1}{\alpha