28 problems

Let $(\alpha,A_E)$ be the Schrödinger cocycle associated with a one-dimensional analytic quasi-periodic Schrödinger operator, where…

Aubry–André conjecture. The Lebesgue measure of the spectrum for any irrational frequency is

Dry ten Martini conjecture. The spectrum of the almost Mathieu operator is a Cantor set and all allowed gaps are present.

Let $\theta\in\mathbb{R}$ be completely resonant with respect to a frequency $\alpha$, meaning that $2\theta\in\alpha\mathbb{Z}+\mathbb{Z}$. For the almost Mathieu operator…

Let $\mathbb{T}^2$ be the two-dimensional torus, and consider a quasiperiodic Schrödinger operator on $\mathbb{Z}$ with a smooth potential sampled along a shift on $\mathbb{T}^2$.…

Critical energy conjecture.

Let $\mathcal N(E)$ denote the integrated density of states of a one-dimensional analytic quasi-periodic Schrödinger operator, and let $\omega(E)$ denote its acceleration. The Höld…

Let $\alpha$ be irrational and let $v\in C^\omega(\mathbb{T},\mathbb{R})$. Write $\Sigma_{v,\alpha}$ for the spectrum and $N_{v,\alpha}$ for the integrated density of states. An en…

Let $p$ and $q$ be coprime integers with $q$ odd, and let $\Sigma_{p/q}$ denote the periodic almost Mathieu spectrum. Let $C=0.915\ldots$ be Catalan's constant. Thouless-type spect…

Let $H(x)$ be an analytic quasiperiodic Schrödinger operator, with frequency $\alpha$ and energy-dependent Lyapunov exponent $L(E)$, and let $\beta(\alpha)$ be the measure of Liouv…

Let $f$ be a potential defining either the Maryland model or the almost Mathieu model, let $\beta(\alpha)$ denote the measure of Liouvillenness of the frequency $\alpha$, and let…

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

A Cantorval is a compact subset of the real line with dense interior such that none of its connected components is isolated. Consider a discrete two-dimensional Schrödinger operato…

Let $\alpha$ be irrational, let $v\in C^\omega(\mathbb T,\mathbb R)$, and let $\Sigma_{v,\alpha}$ be the spectrum with integrated density of states $N_{v,\alpha}$. A type I energy…

Spectral transition conjecture. For $\lambda\in\mathcal{R}_1\cap\mathcal{R}_s$, the operator $H^{\mathrm{EHM}}_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum if

Resonance competition conjecture. For the almost Mathieu family, frequency resonances and phase resonances are the only types of resonances that appear, and the competition between…

Let the almost-Mathieu operator be the one-dimensional nearest-neighbor hopping lattice model with a cosine potential, and let its associated cocycle be the corresponding dynamical…

Let the almost Mathieu potential be $v(\theta)=\cos (2\pi \theta)$. Consider the corresponding two-particle quasiperiodic model with an interaction potential. FIKS conjecture. Ther…

Let the potential be a general even analytic function, let the frequency be chosen from a full-measure set, and let $L(E)$ denote the Lyapunov exponent at energy $E$. Exponentially…

For the almost Mathieu family, let $\alpha$ be the frequency, let $\theta$ be the phase, let $\beta(\alpha)$ measure the strength of frequency resonances, and let…

Localization conjecture. For almost every phase $x$, the almost Mathieu family exhibits localization whenever

Aubry–André phase transition conjecture. The critical line separating purely singular continuous spectrum from pure point spectrum is

Consider a quasiperiodic Schrödinger operator with analytic or Lipschitz potential, including the singular-potential family discussed in the source, and let $L(E)$ denote its Lyapu…

The arithmetic phase-transition conjecture. The value $\lambda=e^\gamma$ should be the phase transition point from singular continuous spectrum to pure point spectrum for Diophanti…

Let $\alpha\in\mathbb R\setminus\mathbb Q$, let $p_n/q_n$ be its continued-fraction approximants, and define … For the almost Mathieu operator … call $\theta$ $\alpha$-Diophantine…