Resonance-free regions for diffractive trapping by conormal potentials

We consider the Schr"odinger operator $$P=h^2{\mathrm{\Delta }}_g+V$$ on ${\mathbb{R}}^n$ equipped with a metric $g$ that is Euclidean outside a compact set. The real-valued potential $V$ is assumed to be compactly supported and smooth except at conormal singularities of order $-1-\alpha$ along a compact hypersurface $Y\mathrm{.}$ For $\alpha >2$ (or even $\alpha >1$ if the classical flow is unique), we show that if $E_0$ is a non-trapping energy for the classical flow, then the operator $P$ has no resonances in a region $$[E_0-\delta ,E_0+\delta ]-i[0,{\nu }_{0}h\log (1\mathrm{/}h)]\mathrm{.}$$ The constant ${\nu }_0$ is explicit in terms of $\alpha$ and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line.