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 $\varphi ( P - z )^{- 1} \varphi$, for any $\varphi \in C^{\infty}_{0} ( \mathbb R^{n} )$, by the norm of ${\mathds{1}}_{\mathcal C_{a,b}} ( P - z )^{- 1} {\mathds{1}}_{\mathcal C_{a,b}}$ where $\mathcal C_{a,b} = \{ x \in \mathbb R^{n} ; \ a < \vert x \vert < b \}$ and $a \gg 1$. For $z$ in the unphysical sheet with $- M h \vert \ln h \vert \leq \operatorname{Im} z \leq 0$, we prove that this estimate holds with a constant $\frac{h}{\vert \operatorname{Im} z \vert} e^{C \vert \operatorname{Im} z \vert / h}$. We also study the resonant states $u$ of the operator $P$ and we obtain bounds for $\Vert \varphi u \Vert$ by $\Vert \mathds{1}_{\mathcal C_{a,b}} u \Vert$. These results hold without any assumption on the trapped set nor any assumption on the multiplicity of the resonances.