Spectral problems in open quantum chaos
Spectral problems in open quantum chaos
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开放量子混沌中的谱问题
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
S. Nonnenmacher
中科院分区：
文献类型：
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作者：
S. Nonnenmacher
我们提出了一个概述的数学结果和方法的光谱研究的半经典薛定谔（或波）运营商的散射系统，在相应的经典动力学是混沌的情况下，更准确地说，我们假设，在一定的能量范围内，经典的哈密顿流承认一个分形集的陷阱轨迹，主机混沌（双曲）动力学。我们的目标是将这个被困集的信息与半经典极限中量子共振的分布联系起来。我们的研究包括几个共享这些动力学特性的模型：自由运动外凸硬障碍物的工会，散射某些家庭的compensated支持的潜力，几何散射流形（恒定或可变）负曲率。我们还考虑了开放量子映射的玩具模型，并绘制了与散射流的庞加莱截面相关的量子单值算子的构造。长寿命共振的半经典密度表现出分形外尔定律，这与相应的亚稳态由分形捕获集（及其输出尾）“支持”的事实有关。我们还描述了一个经典的共振谱中的间隙的存在下，相当于一个统一的量子衰减率的下限条件，并提出了一个证明这个差距在一个相当一般的情况下，使用量子monodromy运营商。
We present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrödinger (or wave) operators of scattering systems, in cases where the corresponding classical dynamics is chaotic; more precisely, we assume that in some energy range, the classical Hamiltonian flow admits a fractal set of trapped trajectories, which hosts chaotic (hyperbolic) dynamics. The aim is then to connect the information on this trapped set with the distribution of quantum resonances in the semiclassical limit. Our study encompasses several models sharing these dynamical characteristics: free motion outside a union of convex hard obstacles, scattering by certain families of compactly supported potentials, geometric scattering on manifolds with (constant or variable) negative curvature. We also consider the toy model of open quantum maps, and sketch the construction of quantum monodromy operators associated with a Poincaré section for a scattering flow. The semiclassical density of long-living resonances exhibits a fractal Weyl law, related to the fact that the corresponding metastable states are ‘supported’ by the fractal trapped set (and its outgoing tail). We also describe a classical condition for the presence of a gap in the resonance spectrum, equivalently a uniform lower bound on the quantum decay rates, and present a proof of this gap in a rather general situation, using quantum monodromy operators.