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Symplectic Normal Form and Growth of Sobolev Norm

Zhenguo Liang, Jiawen Luo, Zhiyan Zhao

Abstract

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schrödinger representations. Each pattern of Sobolev norm behavior corresponds to a specific $n-$dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2. When applied to periodically or quasi-periodically forced $n-$dimensional quantum harmonic oscillators, we identify novel growth rates for the $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{λst}$ (with $λ>0$) and $t^{(2n-1)s}+ ιt^{2ns}$ (with $ι\geq 0$). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3. As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the ``fastest" growth, as articulated in Theorem 1.4.

Symplectic Normal Form and Growth of Sobolev Norm

Abstract

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schrödinger representations. Each pattern of Sobolev norm behavior corresponds to a specific dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2. When applied to periodically or quasi-periodically forced dimensional quantum harmonic oscillators, we identify novel growth rates for the norm as tends to infinity, such as (with ) and (with ). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3. As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the ``fastest" growth, as articulated in Theorem 1.4.
Paper Structure (29 sections, 37 theorems, 247 equations)

This paper contains 29 sections, 37 theorems, 247 equations.

Table of Contents

  1. Introduction of the Main Results
  2. Reducibility and symplectic normal forms
  3. Classification of growth rates (GRs) of Sobolev norms
  4. Strategy of proof
  5. Index of notations
  6. Description of the remaining of paper
  7. Preliminaries
  8. Sobolev space and Sobolev norm
  9. Schrödinger representation
  10. Metaplectic representation
  11. Bounds of Sobolev norms under the representations
  12. Bounds of Sobolev norms under Metaplectic representation
  13. Bounds for symplectic matrix with diagonal block form
  14. Elementary symplectic (ES) matrices
  15. Proof of (\ref{['equiMeta']}) for $s>0$.
  16. ...and 14 more sections

Key Result

Theorem 1.1

(Reducibility and classification of QNFs) If the system $z'={\mathcal{A}}(t)z+\ell(t)$, with ${\mathcal{A}}(\cdot)\in C_b^0({\Bbb R},{\rm sp}(n,{\Bbb R}))$ and $\ell(\cdot)\in C_b^0({\Bbb R}, {\Bbb R}^{2n})$ as in Eq. (orig-equ-1), is reducible, then Eq. (orig-equ-1) is also reducible, via some $L^2

Theorems & Definitions (61)

  • Definition 1
  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Remark 3
  • Theorem 1.2
  • Remark 4
  • Theorem 1.3
  • Theorem 1.4
  • Remark 5
  • ...and 51 more