A doubled Gordon threshold for palindromic quasiperiodic Schrödinger operators

Abstract

We consider one-frequency quasiperiodic Schrödinger operators \[ (H_{v,α,θ}u)(n) = u(n+1)+u(n-1) + v(θ+nα)u(n) \] acting on ${\ell }^2(\mathbb{Z})$, where $\hat{I}\pm \notin \mathbb{Q}$ and $v\in C^2(\mathbb{T},\mathbb{R})$ is an even function. We develop a new Gordon-type method that exploits approximate repetitions and palindromic symmetries simultaneously. Denote by $L(E)$ the Lyapunov exponent and let \[β(α) = \limsup_{|k|\to\infty} -\frac{\log\|kα\|_{\mathbb R/\mathbb Z}}{|k|}. \] We prove that, for every completely resonant phase $2\hat{I}¸\in \hat{I}\pm \mathbb{Z}+\mathbb{Z}$, $E$ cannot be an eigenvalue if $L(E)<2{\hat{I}}^2(\hat{I}\pm )$. As an application, consider the almost Mathieu operator \[ (H_{λ,α,θ}u)(n) = u(n+1)+u(n-1) + 2λ\cos\bigl(2π(θ+nα)\bigr)u(n). \] We show that if $2\hat{I}¸\in \hat{I}\pm \mathbb{Z}+\mathbb{Z}$ and $1<|\hat{I}»|<e^{2{\hat{I}}^2(\hat{I}\pm )}$, then $H_{\hat{I}»,\hat{I}\pm ,\hat{I}¸}$ has purely singular continuous spectrum. This resolves the remaining absence-of-eigenvalues part of a conjecture of Avila and Jitomirskaya concerning the sharp spectral transition for completely resonant phases.