We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator.

We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schrödinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential.

For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them.Let M = Γ\H n , n ≥ 2, be a convex cocompact quotient of hyperbolic space, i.e. a conformally compact manifold of constant negative curvature. Let δ Γ ∈ [0, n − 1) be the Hausdorff dimension of its limit set, which by Patterson-Sullivan theory equals the abscissa of convergence of its Poincaré series [Pa,Su79]. Let Z Γ (s) be the Selberg zeta function: Z Γ (s) = exp − γ∈P ∞ m=1 1 arXiv:1206.2255v5 [math.AP]

Abstract. We study the Gross-Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit.

We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two.