Resolvent Estimates for Normally Hyperbolic Trapped Sets

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable – see §1.2 – and our motivation comes partly from considering the wave equation for slowly rotating Kerr black holes, whose trapped photon spheres have precisely that dynamical structure – see §2. From the semiclassical point of view an example to keep in mind is given by P (z) = −h∆ + V (x)− 1− z , V ∈ C∞ c (R;R) ,