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 \(\exp \left( Ch^{-4/(\alpha +3)}\right) \).