  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 $(-h^2\Delta + V-\lambda^2)^{-1}$ as
an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance
$\lambda$ with real part in a compact interval away from $0$ has imaginary part
at most
  \begin{equation*}
  \mathrm{Im} \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 $-\Delta u = Vu$ 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*}
