Semiclassical resolvent bounds for weakly decaying potentials

Abstract

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

15 pages