Sharp resolvent bounds and resonance-free regions

2017/05/18 by Ingremeau, Maxime
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1705.06548

Abstract

In this note, we consider semiclassical scattering on a manifold which is Euclidean near infinity or asymptotically hyperbolic. We show that, if the cut-off resolvent satisfies polynomial estimates in a strip of size O(h |log h|^-α) below the real axis, for some α≥ 0, then the cut-off resolvent is actually bounded by O(|log h|^α+1 h^-1) in this strip. As an application, we improve slightly the estimates on the real axis given by Bourgain and Dyatlov in the case of convex co-compact surfaces.

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