Resonance-free regions for diffractive trapping by conormal potentials

We consider the Schrodinger operator [ P=h^2 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 $Y.$ 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, u_0 h log(1/h)]. ] The constant $ u_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.