Semiclassical resolvent estimates for bounded potentials

We study the cut-off resolvent of semiclassical Schr{"o}dinger operators on ${\mathbb{R}}^d$ with bounded compactly supported potentials $V$. We prove that for real energies ${\lambda }^2$ in a compact interval in ${\mathbb{R}}_{+}$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} | \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi |{L^2\to H^1} \leq C ,$\mathrm{e}$^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\mathrm{\Delta }+V-{\lambda }^2{)}^{-1}$ as an operator $L_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^2\to H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} $\mathrm{I}\mathrm{m}$ \lambda \leq - C^{-1} ,$\mathrm{e}$^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\mathrm{\Delta }u=Vu$ with $0\not\equiv V\in L^{\infty}($\mathbb{R}$^d)$\mathrm{.}Weshowthatthereexistaconstant$M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} $\mathrm{e}$^{-M |V|_{\infty}^{2/3} R^{4/3}}|u|^2_2. \end{equation*}