Resonance-free regions for diffractive trapping by conormal potentials

We consider the Schr\"odinger 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 $$[E_0 - \delta, E_0 + \delta] - i[0,\nu_0 h \log(1/h)].$$ 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.