AI 中文总结
研究单频拟周期薛定谔算子，开发新戈登型方法，利用近似重复和回文对称性。证明对于完全共振相位，若李雅普诺夫指数\(L(E)<2\beta(\alpha)\)，\(E\)不是特征值。应用于几乎马修算子，解决相关猜想的无特征值部分。
AI 中文摘要
我们考虑作用于\(\ell^2(\mathbb Z)\)的单频拟周期薛定谔算子\((H_{v,\alpha,\theta}u)(n)=u(n + 1)+u(n - 1)+v(\theta + n\alpha)u(n)\)，其中\(\alpha\notin\mathbb Q\)且\(v\in C^2(\mathbb T,\mathbb R)\)是偶函数。我们开发了一种同时利用近似重复和回文对称性的新戈登型方法。用\(L(E)\)表示李雅普诺夫指数，令\(\beta(\alpha)=\limsup_{|k|\to\infty}-\frac{\log\|k\alpha\|_{\mathbb R/\mathbb Z}}{|k|}\)。我们证明，对于每个完全共振相位\(2\theta\in\alpha\mathbb Z+\mathbb Z\)，若\(L(E)<2\beta(\alpha)\)，则\(E\)不是特征值。作为应用，考虑几乎马修算子\((H_{\lambda,\alpha,\theta}u)(n)=u(n + 1)+u(n - 1)+2\lambda\cos(2\pi(\theta + n\alpha))u(n)\)。我们表明，若\(2\theta\in\alpha\mathbb Z+\mathbb Z\)且\(1<|\lambda|<e^{2\beta(\alpha)}\)，则\(H_{\lambda,\alpha,\theta}\)具有纯奇异连续谱。这解决了阿维拉和吉托米尔斯卡娅关于完全共振相位尖锐谱跃迁猜想中剩余的无特征值部分。
英文摘要
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 $α\notin\mathbb Q$ 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 $2θ\inα\mathbb Z+\mathbb Z$, $E$ cannot be an eigenvalue if $L(E)<2β(α)$. 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 $2θ\inα\mathbb Z+\mathbb Z$ and $1<|λ|<e^{2β(α)}$, then $H_{λ,α,θ}$ 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.
