Maciej Zworski

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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)
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)
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)
  • Patrick G, S Alinhac, L Baratchart, T Kap-Peler, S Kuksin, J Leblond +2 others
  • 2009
On considère l'´ equation hamiltonienne suivante sur l'espace de Hardy du cercle i∂ t u = Π(|u| 2 u) , o` u Π désigne le projecteur de Szegö. Cetté equation est un cas modèle d'´ equation sans aucune propriété dispersive. Onétablit qu'elle admet une paire de Lax et une infinité de lois de conservation en involution, et qu'elle peutêtre approchée par une(More)
For scattering on the modular surface and on the hyperbolic cylinder, we show that the solutions of the wave equations can be expanded in terms of resonances, despite the presence of trapping. Expansions of this type are expected to hold in greater generality but have been understood only in non-trapping situations. 1. Introduction In this note we give two(More)
Resonances, or scattering poles, are complex numbers which mathematically describe meta-stable states: the real part of a resonance gives the rest energy, and its imaginary part, the rate of decay of a meta-stable state. This description emphasizes the quantum mechanical aspects of this concept but similar models appear in many branches of physics,(More)
A well known result of Jaffard states that an arbitrary region on a torus controls , in the L 2 sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C ∞ .
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)
  • Johannes Sj¨ostrand, Maciej Zworski, J Sj¨ostrand, M Zworski
  • 2008
Trace formulae provide one of the most elegant descriptions of the classical-quantum correspondence. One side of a formula is given by a trace of a quantum object, typically derived from a quantum Hamiltonian, and the other side is described in terms of closed orbits of the corresponding classical Hamiltonian. In algebraic situations, such as the original(More)