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Non-resonant conditions for the Klein-Gordon equation on the circle

Roberto Feola, Jessica Elisa Massetti

Abstract

We consider the infinite dimensional vector of frequencies $ω(m)=( \sqrt{j^2+m})_{j\in \mathbb{Z}}$, $m\in [1,2]$ arising form a linear Klein-Gordon equation on the one dimensional torus and prove that there exists a positive measure set of masses $m'$s for which $ω(m)$ satisfies a diophantine condition similar to the one introduced by Bourgain in (JFA, 2005), in the context of Schrödinger equation with convolution potential. The main difficulties we have to deal with are the asymptotically linear nature of the (infinitely many) $ω_{j}'$s and the degeneracy coming from having only one parameter at disposal for their modulation. As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category.

Non-resonant conditions for the Klein-Gordon equation on the circle

Abstract

We consider the infinite dimensional vector of frequencies , arising form a linear Klein-Gordon equation on the one dimensional torus and prove that there exists a positive measure set of masses s for which satisfies a diophantine condition similar to the one introduced by Bourgain in (JFA, 2005), in the context of Schrödinger equation with convolution potential. The main difficulties we have to deal with are the asymptotically linear nature of the (infinitely many) s and the degeneracy coming from having only one parameter at disposal for their modulation. As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category.
Paper Structure (8 sections, 9 theorems, 146 equations)

This paper contains 8 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. The problem of small divisors.
  3. Small divisors
  4. A weak-diophantine condition for the Klein-Gordon
  5. Improved estimates for the Klein-Gordon equation
  6. Homological equation
  7. Functional setting and homogeneous Hamiltonians
  8. Inversion of the adjoint action

Key Result

Theorem 1.2

(Measure estimates for the Klein-Gordon). There exists a positive measure set $\mathfrak{Q}\subseteq [1,2]$ such that for any $\mathtt{m}\in\mathfrak{Q}$, the vector $\omega(\mathtt{m})$ satisfies the following: for any $\ell=\alpha-\beta$, $(\alpha,\beta)\in \Lambda$ one has where $\tau(\ell):={\mathtt{d}}(\ell)({\mathtt{d}}(\ell)+4)$, and $\mathtt{C}$ is a positive pure constant large enough. M

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • proof : Proof of Proposition \ref{['thm:main']}
  • ...and 13 more