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Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models

Yujia An, Xifeng Su

TL;DR

The paper analyzes phonon gaps for equilibria of perturbed almost-periodic Frenkel-Kontorova models. It contrasts perturbative KAM equilibria, which lack a phonon gap and can slide, with equilibria arising from anti-integrable limits, which exhibit a phonon gap and pinning. Using an operator-theoretic framework, it shows that KAM equilibria have a zero mode in the Hessian spectrum, while anti-integrable-limit equilibria become spectrally gapped for sufficiently large coupling $\lambda$, with an explicit lower bound on the gap. The results clarify how near-integrable versus strongly perturbed regimes influence hyperbolicity and energy barriers in quasi-periodic FK systems, with implications for transport and stability in quasi-crystal-like media.

Abstract

For generalized Frenkel-Kontorova models subjected to almost-periodic media, employing both the KAM method and the approach of `anti-integrable' limits, two different types of equilibria are obtained in \cite{an2024kamtheoryalmostperiodicequilibria} and \cite{du2024anti} respectively. We study the phonon gap around these equilibria and we find that the KAM equilibria do not have a phonon gap but the equilibria obtained by anti-integrable limits do have.

Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models

TL;DR

The paper analyzes phonon gaps for equilibria of perturbed almost-periodic Frenkel-Kontorova models. It contrasts perturbative KAM equilibria, which lack a phonon gap and can slide, with equilibria arising from anti-integrable limits, which exhibit a phonon gap and pinning. Using an operator-theoretic framework, it shows that KAM equilibria have a zero mode in the Hessian spectrum, while anti-integrable-limit equilibria become spectrally gapped for sufficiently large coupling , with an explicit lower bound on the gap. The results clarify how near-integrable versus strongly perturbed regimes influence hyperbolicity and energy barriers in quasi-periodic FK systems, with implications for transport and stability in quasi-crystal-like media.

Abstract

For generalized Frenkel-Kontorova models subjected to almost-periodic media, employing both the KAM method and the approach of `anti-integrable' limits, two different types of equilibria are obtained in \cite{an2024kamtheoryalmostperiodicequilibria} and \cite{du2024anti} respectively. We study the phonon gap around these equilibria and we find that the KAM equilibria do not have a phonon gap but the equilibria obtained by anti-integrable limits do have.
Paper Structure (4 sections, 4 theorems, 21 equations)

This paper contains 4 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The phonon gap
  3. KAM equilibria without a phonon gap
  4. Equilibria by anti-integrable limits with a phonon gap

Key Result

Theorem 1

Let $h(\theta)=\theta+\tilde{h}(\theta)$, $\tilde{h}(\theta)=\hat{h}(\alpha\theta)=\sum_{k\in\mathbb Z^\mathbb{N}_*}\hat{h}_ke^{ik\cdot\alpha\theta}$, $\hat{h}_0=0$, $\hat{h}\in\mathscr A_{\rho_0}^1$. $\alpha\in[0,1]^{\mathbb N}$ is rationally independent. $\partial_{\alpha}\widehat{V}\in\mathscr A_ If $\|\mathcal{E}[\hat{h}]\|_{\rho_0}$ is small enough $(\|\mathcal{E}[\hat{h}]\|_{\rho_0}\le\epsil

Theorems & Definitions (17)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 2
  • Theorem 1: an2024kamtheoryalmostperiodicequilibria
  • Remark 3
  • Theorem 2
  • ...and 7 more