Quantum decay rates in chaotic scattering

By S. Nonnenmacher, Maciej Zworski. 2007

173 citations recorded by the source corpus.

We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/869265dd578c228f8565e2d5bb0d6076a064364a

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Citations in the corpus

Strichartz Estimates on Schwarzschild Black Hole Backgrounds (2008, background): “The logarithmic loss is not surprising, since it is characteristic of geometries with trapped hyperbolic orbits (see for instance [9], [12], [34]).”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, background): “To see that large energy estimates also hold for (Pσ − ıQσ)−1 near R, namely in Imσ > −C, one considers (Pσ − ı(Qσ + Q′σ))−1, which enjoys such estimates but is not holomorphic, and then use the semiclassical resolvent gluing of Datchev-Vasy [15] together with the semiclassical normally hyperbolic t”
Restriction and spectral multiplier theorems on asymptotically conic manifolds (2010, unknown): “• Prove (or disprove) the restriction theorem for high energies in the presence of trapping, in the case that the trapped set is hyperbolic and the topological pressure assumption of [Nonnenmacher and Zworski 2009] and [Burq et al.”
The global non-linear stability of the Kerr-de Sitter family of black holes (2016, result): “These works in turn rely crucially on the breakthrough work of Wunsch–Zworski [WZ11] on estimates at normally hyperbolically trapped sets, later extended and simplified by Nonnenmacher–Zworski [NZ09] and Dyatlov [Dya14]; see also [HV14].”