Quantum decay rates in chaotic scattering

• 成果类型: Article
• 署名作者: Nonnenmacher, Stephane; Zworski, Maciej
• 署名单位: CEA; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; University of California System; University of California Berkeley
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-009-0041-z
• 发表日期: 2009
• 页码: 149-233
• 关键词: resonances; MANIFOLDS; laplacian; resolvent; density; bounds; Axiom

摘要

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.