EVENT DETAILS AND ABSTRACT

Title: Semi-classical estimates for the cut-off resolvent and the resonance states for trapping perturbations
Speaker: Vesselin Petkov
Speaker Info: Bordeaux
Brief Description:
Special Note: Note unusual date and time
Abstract:

We study the semi-classical estimates for the cut-of resolvent $R_{\varphi}(z, h) =\varphi (P(h) - z)^{-1} \varphi$ of a semi-classical operator $P(h).$ For real $z$ these estimates depend on the support of the cut-off $\varphi \in C_0^{\infty}(\R^n)$ and if the cut-off is equal to 1 on the trapped set, $R_{\varphi}(z, h)$ could grow exponentially as $h$ goes to 0, while for cut-off supported sufficiently far from the trapped set we have $R_{\varphi}(z, h) = {\mathcal O}(1/h)$. In this talk we show that the situation is different if $z \in \C$ and we prove that both cut-off resolvents are related by an inequality with a constant depending on $\frac{1}{|\Im z|}$. Our results hold without any assumptions on the trapping perturbation. We examine also the estimates of the resonance states $u$ determined as the solutions of $(P(h) - z)u = 0$ and we show that $\|\varphi u\|$ can be estimated by ${\mathcal O}(\frac{h}{|\Im z|})\|\one_{a \leq |x| \leq b} u\|$. This is a work in collaboration with J. F. Bony.