Semiclassical resolvent estimates for bounded potentials

Abstract

We study the cut-off resolvent of semiclassical Schr{ö}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*} \| χ(-h^2Δ+ V-λ^2)^{-1} χ\|_{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 $(-h^2\Delta +V-{\lambda }^2{)}^{-1}$ as an operator $L_{\mathrm{comp}}^2\to H_{\mathrm{loc}}^2$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} λ\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 $-\Delta u=Vu$ with $0\not\equiv V\in L^{\infty }({\mathbb{R}}^d)$. We show that there exist a constant $M>0$ 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*}

21 pages