Resonance-free regions for diffractive trapping by conormal potentials

We consider the Schrödinger operator P = h²Δ_g + V on Rⁿ 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 − α along a compact hypersurface Y .Forα>2(orevenα>1 if the classical flow is unique), we show that if E₀ is a non-trapping energy for the classical flow, then the operator P has no resonances in a region [_E0 − δ,E₀ + δ^] − i^[0,ν₀hlog(1/h)^]. The constant ν₀ is explicit in terms of α 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.