3 problems

Let $H_\theta$ denote the almost Mathieu operator at coupling parameter $\lambda=1$, and let $\theta\in[0,1]$ be the frequency parameter. Béguin–Valette–Zuk norm conjecture. … The…

Sharp arithmetic transition conjecture. For (arithmetically) almost every phase, frequency resonances are the only type of resonance that appears. The spectrum is pure point when

Let the full butterfly be the parameterized spectrum of the almost Mathieu operators, including irrational values of the vertical parameter $\theta$, and let $S=S_{M,r,\pm}$ be a s…