Spectral Gaps of Almost Mathieu Operator in Exponential Regime

For almost Mathieu operator $(H_{lambda,alpha, heta}u)_n=u_{n+1}+u_{n-1}+2lambda cos2pi( heta+nalpha)u_n$, the dry version of Ten Martini problem predicts that the spectrum $Sigma_{lambda,alpha}$ of $ H_{lambda,alpha, heta}$ has all gaps open for all $lambda
eq 0$ and $ alpha in mathbb{R}\backslash mathbb{Q}$. Avila and Jitomirskaya prove that $Sigma_{lambda,alpha}$ has all gaps open for Diophantine $alpha$ and $0<|lambda|<1$. In the present paper, we show that $Sigma_{lambda,alpha}$ has all gaps open for all $ alpha in mathbb{R}\backslash mathbb{Q}$ with small $lambda$.

Spectral Gaps of Almost Mathieu Operator in Exponential Regime Institutional Repository Document