Improved resolvent bounds for radial potentials. II

Abstract

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