Analytic quasi-periodic Schrödinger operators and rational frequency approximants (Q1930905)

Let \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\). Consider a discrete quasi-periodic Schrödinger operator \(H_{\alpha,\theta}\) acting on \(l^2(\mathbb{Z})\): \[ (H_{\alpha,\theta}f)_n:=f_{n-1}+f_{n+1}+v(\alpha n+\theta)f_n,\quad n\in\mathbb{Z}. \] It is assumed that \(v\) is an analytic function, \(v\in C^{\omega}(\mathbb{T},\mathbb{R})\), the frequency \(\alpha\) is irrational, and the phase \(\theta \in \mathbb{T}\). It is known that the spectrum of \(H_{\alpha,\theta}\) and its ac-part do not depend on \(\theta\): \[ \sigma(H_{\alpha,\theta})=\Sigma(\alpha),\qquad \sigma_{ac}(H_{\alpha,\theta})=\Sigma_{ac}(\alpha)\quad \text{ for all } \theta\in\mathbb{T}. \] For \(\beta\in\mathbb{R}\), define the sets \[ S_+(\beta):=\bigcup_{\theta\in\mathbb{T}}\sigma(H_{\beta,\theta}),\qquad S_-(\beta):=\bigcap_{\theta\in\mathbb{T}}\sigma_{ac}(H_{\beta,\theta}). \] Notice that \(S_+(\beta)=\Sigma(\beta)\) and \(S_-(\beta)=\Sigma_{ac}(\beta)\) if \(\beta\) is irrational. The central result of the paper states that, for any irrational frequency \(\alpha\), there is a sequence of rationals \(p_k/q_k\to \alpha\) such that \[ \lim_{k\to\infty} S_-\Big(\frac{p_k}{q_k}\Big)=S_-(\alpha)=\Sigma_{ac}(\alpha) \] and \[ \lim_{k\to\infty} S_+\Big(\frac{p_k}{q_k}\Big)=S_+(\alpha)=\Sigma(\alpha). \] The limit is understood as a convergence of the corresponding characteristic functions almost everywhere (w.r.t. Lebesgue measure). As an immediate corollary, one concludes that \[ \lim_{k\to\infty} \Big|S_-\Big(\frac{p_k}{q_k}\Big)\Big|=|\Sigma_{ac}(\alpha)|,\quad \lim_{k\to\infty} \Big|S_+\Big(\frac{p_k}{q_k}\Big)\Big|=|\Sigma(\alpha)|, \] where \(|S|\) denotes the Lebesgue measure of a set \(S\). Furthermore, it is shown that, for Diophantine \(\alpha\), the sequence \(\frac{p_k}{q_k}\) can be taken as a sequence of canonical continuous fraction approximants. For non-Diophantine \(\alpha\), one should restrict to a subsequence of sufficiently strong approximants. Namely, if \(\alpha\) is non-Diophantine and \(\frac{p_k}{q_k}\) are canonical continuous fraction approximants, then, for any \(r>1\), \[ q_{k+1}>q_k^r\quad \text{infinitely often}. \] It is precisely these subsequences for which convergence takes place.