2004/12/30 by Mikhail Karasev, Karasev, Mikhail
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum optics and atomic interactions #math-ph #math.MP #math.QA #msc:34E20 #msc:35Lxx #msc:81Qxx
paper · pdf · doi:10.48550/arxiv.math/0412542
Latex, 60pages, Part II added, minor corrections to Part I
arxiv created 2005/05/27 · arxiv updated 2009/12/01
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.