Resonance-free regions for diffractive trapping by conormal potentials

Abstract

We consider the Schrödinger operator \[ P=h^2 Δ_g + V \] 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 \[ [E_0 - δ, E_0 + δ] - i[0,ν_0 h \log(1/h)]. \] 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.

20 pages; added Section 2.4 on applications to quantum evolution