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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… (More)

- Stéphane Nonnenmacher, Maciej Zworski
- 2007

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)

- Richard Melrose, Maciej Zworski
- 1996

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)

- Andr´as Vasy, Maciej Zworski
- 2007

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)

Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only elementary calculus. In the case of the famous Bunimovich billiard table shown in Fig.1 we prove that the wave functions have… (More)

- Justin Holmer, Maciej Zworski
- 2007

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)

- Justin Holmer, Maciej Zworski
- 2007

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)

- Stéphane Nonnenmacher, Maciej Zworski
- 2009

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)

- Maciej Zworski
- 2015

We revisit Vasy's method [Va1],[Va2] for showing meromorphy of the resolvent for (even) asymptotically hyperbolic manifolds. It provides an effective definition of resonances in that setting by identifying them with poles of inverses of a family of Fredholm differential operators. In the Euclidean case the method of complex scaling made this available since… (More)