Semiclassical estimates of the cut-off resolvent for trapping perturbations

Abstract

This paper is devoted to the study of the cut-off resolvent of a semiclassical “black box” operator $P$. We estimate the norm of $\phi (P-z{)}^{-1}\phi$, for any $\phi \in C_0^{\infty }({\mathbb{R}}^n)$, by the norm of ${11}_{{\mathcal{C}}_{a,b}}(P-z{)}^{-1}{11}_{{\mathcal{C}}_{a,b}}$ where ${\mathcal{C}}_{a,b}=\{x\in {\mathbb{R}}^n;a<|x|<b\}$ and $a\gg 1$. For $z$ in the unphysical sheet with $-Mh|\ln h|\le \mathrm{Im}z\le 0$, we prove that this estimate holds with a constant $\frac{h}{|\mathrm{Im}z|}{e}^{C|\mathrm{Im}z|/h}$. We also study the resonant states $u$ of the operator $P$ and we obtain bounds for $\Vert \phi u\Vert$ by $\Vert 11_{{\mathcal{C}}_{a,b}}u\Vert$. These results hold without any assumption on the trapped set nor any assumption on the multiplicity of the resonances.