非对易代数、纳米结构和由共振产生的量子动力学
Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances
摘要：
我们观察了相空间中共振平衡点和共振单变子流形的“量子”性质。Birkhoff或Floquet-Lyapunov频率之间的共振产生具有多项式对易关系的量子代数。这些代数的不可约表示和相干态对应于经典共振运动附近的某种量子纳米结构。基于这种表示论和纳米几何，我们描述了薛定谔方程或波型方程在各种制度和区域（直到量子混沌边界）的共振谱和长时间渐近性，共振局域化和聚焦，共振绝热和类自旋效应。我们讨论了如何数学相空间纳米结构涉及到物理纳米级物体，如点，量子线等，我们还表明，即使在物理宏观亥姆霍兹通道的共振意味着一个特定的量子特性的经典波传播。
英文摘要：
We observe ``quantum'' properties of resonance equilibrium points and resonance univariant submanifolds in the phase space. Resonances between Birkhoff or Floquet--Lyapunov frequencies generate quantum algebras with polynomial commutation relations. Irreducible representations and coherent states of these algebras correspond to certain quantum nano-structure near the classical resonance motion. Based on this representation theory and nano-geometry, for equations of Schrödinger or wave type in various regimes and zones (up to quantum chaos borders) we describe the resonance spectral and long-time asymptotics, resonance localization and focusing, resonance adiabatic and spin-like effects. We discuss how the mathematical phase space nano-structures relate to physical nanoscale objects like dots, quantum wires, etc. We also demonstrate that even in physically macroscale Helmholtz channels the resonance implies a specific quantum character of classical wave propagation.
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