Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies (Q2301858)

A discrete quasi-periodic Schrödinger operator on \(\ell^2(\mathbb Z)\) is given by \[ (H_{\lambda,f,\alpha,\theta}\,x)_n=x_{n-1}+x_{n+1}+\lambda f(\theta+n\alpha)x_n. \] It depends on four parameters: the potential \(f\), the coupling \(\lambda\), the frequency \(\alpha\), and the phase \(\theta\). The main problem under consideration concerns the upper bounds for the lengths of spectral gaps. It is well known that there is a unique nonzero integer \(m\) associated with each open, spectral gap \(G=(E_m^-, E_m^+)\) in accordance with the value of the fibered rotation number restricted to this gap. Here is the main result on the exponential decay of the lengths of spectral gaps. Theorem. Let the potential \(f\) be analytic on the strip \(\{|\Im z|<h\}\). Then there is an absolute constant \(C\) so that for all frequencies \(\alpha\) with \[ 0\le \beta(\alpha):=\limsup_{k\to\infty}\frac{-\log \|k\alpha\|}{|k|}\le \frac{h}{C}, \ \|a\|:=\min_{n\in\mathbb{Z}} |a-n|, \] there exist \(\lambda_0(f,h,\beta(\alpha))\) and \(m_0(h, \alpha,f,\lambda)\) such that the following bound holds \[ E_m^+-E_m^-\le e^{-\frac{h}{C} |m|}, \ |\lambda|\le\lambda_0, \ |m|\ge m_0. \]