2001/06/25 by Stefan Weigert · 1 citation
Mathematics · Physics and Astronomy · #Floquet theory #Harmonic oscillator #Parametric oscillator #Parametric statistics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Resonance (particle physics) #Spectral Theory in Mathematical Physics #Spectrum (functional analysis) #quant-ph
paper · pdf · doi:10.1088/0305-4470/35/18/312
published as J. Phys. A 35 (2002) 4669 · 10 pages, revtex
arxiv created 2001/06/25 · openalex publication_date 2002/04/26 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically instable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.