Dispersive Estimates for Manifolds with one Trapped Orbit

Abstract

For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: \int_0^T || ρ_s e^{it(Δ_g-V)} u_0 ||_{H^{1/2 - ε}(M)}^2 dt \leq C_T || u_0 ||_{L^2(M)}^2, where $ρ_s(x) \in \Ci(M)$ satisfies $ρ_s = <\dist_g(x,x_0)>^{-s}$, $s> \half$, and $V \in \Ci(M)$, $0 \leq V \leq C$ satisfies $|\nabla V| \leq C <\dist(x,x_0)>^{-1-δ}$ for some $δ>0$. From the local smoothing estimate, we deduce a family of Strichartz-type estimates, which are used to prove two well-posedness results for the nonlinear Schrödinger equation. As a second application, we prove the following exponential local energy decay estimate for solutions to the wave equation when $\dim M = n \geq 3$ is odd and $M$ is equal to $\reals^n$ outside a compact set: \be \int_M |ψ\partial_t u |^2 + | ψ\nabla u |^2 dx} \leq C e^{-t/C} (||u(x,0)||_{H^{1+ε}(M)}^2 + ||D_tu(x,0)||_{H^ε(M)}^2), where $ψ\in \Ci(M)$, $ψ\equiv e^{-|x|^2}$ outside a compact set.