Semiclassical resolvent bounds for weakly decaying potentials

In this note, we prove weighted resolvent estimates for the semiclassical Schr"odinger operator $-h^2 \Delta + V(x) : L^2($\mathbb{R}$^n) \to L^2($\mathbb{R}$^n)$,$n \neq 2$\mathrm{.}Thepotential$V$ is real-valued, and assumed to either decay at infinity or to obey a radial $\alpha$-H"older continuity condition, $0\le \alpha \le 1$, with sufficient decay of the local radial $C^{\alpha }$ norm toward infinity. Note, however, that in the H"older case, the potential need \emph{not} decay. If the dimension $n\ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - \alpha}{3 + \alpha}} [(1-\alpha) \log(h^{-1})+c]\right)$,whilefor$n = 1$itisoftheform$\exp(Ch^{-1})$. A new type of weight and phase function construction allows us to reduce the necessary decay even in the pure $L^{\mathrm{\infty }}$ case.