Improved resolvent bounds for radial potentials

Abstract

We prove semiclassical resolvent estimates for the Schrödinger operator in ${\mathbb {R}}^d$, $d\ge 3$, with real-valued radial potentials $V\in L^\infty ({\mathbb {R}}^d)$. In particular, we show that if $V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right)$ with $\delta >2$, then the resolvent bound is of the form $\exp \left( Ch^{-4/3}\right)$ with some constant $C>0$. We also get resolvent bounds when $1<\delta \le 2$. For slowly decaying $\alpha$—Hölder potentials, we get better resolvent bounds of the form $\exp \left( Ch^{-4/(\alpha +3)}\right)$.