Improved Resolvent Bounds for Radial Potentials

Abstract

We prove semiclassical resolvent estimates for the Schr{\"o}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). In particular, we show that if V (x) = O x --$\delta$ with $\delta$ > 2, then the resolvent bound is of the form e Ch --4/3 with some constant C > 0. We also get resolvent bounds when 1 < $\delta$ $\le$ 2. For slowly decaying $\alpha$-H{\"o}lder potentials we get better resolvent bounds of the form e Ch --4/($\alpha$+3) .