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In this article we prove that for a large class of operators, including Schrödinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary.(More)
Any compact C ∞ manifold with boundary admits a Riemann metric on its interior taking the form x −4 dx 2 + x −2 h near the boundary, where x is a boundary defining function and h is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a(More)
Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply(More)
The purpose of this note is to obtain semiclassical resolvent estimates for long range perturbations of the Laplacian on asymptotically Euclidean manifolds. For an estimate which is uniform in the Planck constant h we need to assume that the energy level is non-trapping. In the high energy limit (that is, when we consider ∆ − λ 2 , as λ → ∞, which is(More)
We study the Gross-Pitaevskii equation with a slowly varying smooth potential , V (x) = W (hx). We show that up to time log(1/h)/h and errors of size h 2 in H 1 , the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, (ξ 2 + sech 2 * V (x))/2. This provides an improvement (h → h 2) compared to previous(More)
We study the Gross-Pitaevskii equation with a delta function potential, qδ 0 , where |q| is small and analyze the solutions for which the initial condition is a soliton with initial velocity v 0. We show that up to time (|q| + v 2 0) − 1 2 log(1/|q|) the bulk of the solution is a soliton evolving according the classical dynamics of a natural effective(More)
We prove resolvent estimates for semiclassical operators such as −h 2 ∆ + V (x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h −M in a strip whose width is determined by a certain topological pressure associated(More)