8 problems

Cantor spectrum conjecture. In general, $\sigma(H_{\theta}^{g})$ should be a Cantor set.

Jitomirskaya's conjecture. For the supercritical almost Mathieu operator, the spectral type—pure point spectrum with localization or singular continuous spectrum—should be complete…

Let $H_{v,\alpha,\theta}$ be a type I operator, and let $L_{v,w,\alpha,\theta}$ be its dual model. The phase parameter is denoted by $\theta$. All-phases absolute-continuity conjec…

Let a one-dimensional quasi-periodic operator have an associated cocycle, and consider the absolutely continuous part of its spectrum. Spectral-cocycle almost-reducibility conjectu…

Let $H_0$ be the Hamiltonian whose potential on the sites $n$ is the quasi-periodic function $v(n)=U\cos(2\pi\omega n)$, where $-2<U<2$ and $\omega$ is irrational. Assume that the…

Let $\alpha$ be irrational and let $\theta$ be any phase. Write $\beta$ and $\delta$ for the arithmetic quantities defined in the paper, and let $\lambda$ be the coupling parameter…

Simon's conjecture. For all $d,v\in\mathbb{N}$ and all $\theta\in\mathbb{T}^{\nu}$, $\sigma_{sing}(H_{\alpha,\theta})$ is constant in $\theta$.

Fix an irrational rotation of the circle and consider the associated one-frequency quasi-periodic Schrödinger operators generated by continuous sampling functions. Generic zero-mea…