Resonance-free regions for diffractive trapping by conormal potentials

TL;DR

This paper establishes resonance-free regions for a Schrödinger operator with conormal singularities, showing how the classical flow's properties influence the absence of resonances near non-trapping energies.

Contribution

It provides explicit resonance-free regions for Schrödinger operators with conormal potentials, extending understanding to piecewise-smooth potentials and linking dynamical properties to spectral gaps.

Findings

Resonance-free region size is optimal for certain piecewise-smooth potentials.

Explicit bounds depend on the order of conormal singularities and classical flow.

No resonances occur in specified regions near non-trapping energies.

Abstract

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.