Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillator

Charbella Abou Khalil

Abstract

Given small initial solutions of the nonlinear quantum harmonic oscillator on $\mathbb{R}$, we are interested in their long time behavior in the energy space which is an adapted Sobolev space. We perturbate the linear part by $V$ taken as multiplicative potentials, in a way that the linear frequencies satisfy a non-resonance condition. More precisely, we prove that for almost all potentials $V$, the low modes of the solution are almost preserved for very long times.

Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillator

Abstract

Given small initial solutions of the nonlinear quantum harmonic oscillator on , we are interested in their long time behavior in the energy space which is an adapted Sobolev space. We perturbate the linear part by taken as multiplicative potentials, in a way that the linear frequencies satisfy a non-resonance condition. More precisely, we prove that for almost all potentials , the low modes of the solution are almost preserved for very long times.
Paper Structure (14 sections, 30 theorems, 217 equations)

This paper contains 14 sections, 30 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Main results and comments
  4. Sketch of the proofs
  5. Notations
  6. Non-resonance condition
  7. Preliminaries on spectral analysis
  8. Non-resonance condition
  9. Hamiltonian formalism
  10. Functional setting
  11. Class of Hamiltonian functions
  12. Birkhoff normal form theorem
  13. Proof of the main result
  14. Appendix

Key Result

Theorem 1.2

Let $N \geq 1,$$r\geq p+1$ arbitrarily large, $\nu >0$ and let $V \in \widehat{H}^1 \cap \mathscr{C}^2$ such that the spectrum of $T+V$ is strongly $N,r$ non-resonant (refer to Definition dep1). Then, there exist $\varepsilon_0>0$ depending on ${\Vert V \Vert}_{\widehat{H}^1}$ and a constant $C>0$ with $2p+2$ being the order of the Hamiltonian non-linearity and $u_j(t) = \int_{\mathbb{R}}u(t,x)\

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 55 more