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Generalized Reducibility and Growth of Sobolev Norms

Zhenguo Liang, Zhiyan Zhao

Abstract

We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed sub-exponential growth rates $f(t)$, either monotone or oscillatory, we explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate $f(t)$.

Generalized Reducibility and Growth of Sobolev Norms

Abstract

We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed sub-exponential growth rates , either monotone or oscillatory, we explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate .
Paper Structure (14 sections, 12 theorems, 110 equations)

This paper contains 14 sections, 12 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Generalized reducibility
  3. Growth of Sobolev norms
  4. Classical-quantum correspondence
  5. Construction for prescribed growth rate
  6. Preliminaries
  7. Schrödinger representation
  8. Metaplectic representation
  9. Conjugation in classical and quantum Hamiltonians
  10. Generalized reducibility - Proof of Theorem \ref{['thm-reduc']}
  11. Growth rate of Sobolev norms
  12. Monotone sub-exponential growth
  13. Oscillatory sub-exponential growth
  14. Proof of (\ref{['gr_lin']})

Key Result

Theorem 1.1

(Generalized reducibility) The quadratic quantum Hamiltonian (orig-equ-1) is reducible in the generalized sense, i.e., there exists an $L^2-$unitary transformation such that the quantum Hamiltonian (orig-equ-1) is conjugated to the constant-coefficient equation

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • ...and 4 more