24 problems

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

Let $\alpha\in{\mathbb R}\setminus{\mathbb Q}$, let $\Sigma_\alpha$ denote the spectrum of the critical almost Mathieu operator $H_{1,\alpha,\theta}$, and let $\dim_H$ denote Hausd…

Let $H_\theta$ be the Harper operator with parameter $\theta$. Ten Martini conjecture. If $\theta$ is irrational, then the spectrum of $H_\theta$ is a Cantor set. The source states…

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

Tang–Kohmoto conjecture. An absolutely continuous spectrum is dominated by points with the trivial scaling index

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…

For the Almost Mathieu operator with irrational frequency $\omega$ and coupling parameter $b$, the spectrum is denoted by $\sigma_b$. A spectral gap is a connected component of the…

Avila–Jitomirskaya's conjecture. If $1<|\lambda|<e^{2\beta(\alpha)}$, then $H_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum; if $|\lambda|>e^{2\beta(\alpha)}$, t…

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 $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-fr…

Let $\operatorname{Sp}_{+}(\alpha,1)$ be the positive half of the critical almost Mathieu spectrum, and let $\operatorname{dim}_{\mathrm{H}}$ denote Hausdorff dimension and…

Let $\operatorname{Sp}_{+}(\alpha,\lambda)$ denote the positive half of the spectrum of the almost Mathieu operator, and let $\operatorname{dim}_{\mathrm{H}}$ denote Hausdorff dime…

Simon's homogeneity conjecture. The spectrum $S(\alpha,\lambda)$ is homogeneous for any $\lambda\neq\pm1$.

Let $S(\alpha,1)$ be the spectrum of the almost Mathieu operator at irrational frequency $\alpha\in\mathbb{R}\setminus\mathbb{Q}$. For $t>0$, let $\widetilde\omega_t$ denote the ga…

Let $P_\ast$ be the fixed point of the period-three renormalization operator, and let ${\text{\teneufm R}}^3$ denote its third iterate. Consider perturbations that change the stren…

Let $P_\ast$ be the fixed point of the period-three renormalization operator, and let $D\operatorname{R}_3(P_\ast)$ have eigenvalue $\mu_2$. Let $\alpha_\ast$ denote the associated…

Almost Mathieu diffusion conjecture. For almost every $\theta,\alpha\in[0,1]$, the equation has a diffusion constant $D(g,\lambda)\in(0,\infty)$ that is smooth for all…

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

Spectral transition conjecture. For an $\alpha$-Diophantine, and hence almost every, phase $\theta$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization—that is, it has onl…

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…

All-phase absence-of-eigenvalues conjecture. The exclusion of the exceptional countable set of phases is unnecessary: the critical almost Mathieu operator has no eigenvalues for ev…

Bellissard's conjecture. There exists some $cbeta\bin(0,1/2]$ such that