Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances
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 Schr\"odinger 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.
引用
@article{arxiv.math/0412542,
title = {Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances},
author = {Mikhail Karasev},
journal= {arXiv preprint arXiv:math/0412542},
year = {2007}
}
备注
Latex, 60pages, Part II added, minor corrections to Part I