14 problems

Simon's conjecture. Cantor spectrum should be a generic phenomenon for one-dimensional almost-periodic operators.

Let $\Lambda_\varphi$ be the one-dimensional Dirac operator on $L^2({\mathbb{R}},{\mathbb{C}}^2)$, with potential $\varphi:{\mathbb{R}}\to{\mathbb{C}}$ satisfying … Call…

Let $f(z)=z^2-c$ with $c>2$, let $J(f)$ be its Julia set, and let $H_{\mu_{J(f)}}$ be the Jacobi matrix associated with the equilibrium measure $\mu_{J(f)}$. A Jacobi matrix is alm…

Equality conjecture. It is conjectured that

Let $\beta$ be an admissible weight on an additive subgroup $\Xi$ of $\mathbb{R}/\mathbb{Z}$ satisfying the GRS-condition, and let $\alpha_1,\alpha_2\geq 0$. Let…

Let $K(\gamma)$ be the Cantor-type set associated with $\gamma$, let $\mu_{K(\gamma)}$ be its associated measure, and let $W_n^2(\mu_{K(\gamma)})$ be the corresponding Widom–Hilber…

Let $K(\gamma)$ be the Cantor-type set determined by $\gamma$, and let $H_{\mu_{K(\gamma)}}$ have recurrence-coefficient sequence $(a_n)_{n=1}^\infty$. An almost periodic sequence…

A self-similar measure is a measure generated by an iterated self-similar construction, including the Cantor measure. A Jacobi matrix is asymptotically almost periodic when its rec…

Let an I.F.S. measure be supported on a Cantor set, and let its associated Jacobi matrix be the tridiagonal matrix determined by the recurrence coefficients of its orthogonal polyn…

For an irrational real number $\theta$, let $H_\theta$ denote Harper's operator, obtained from the image of the operator $x+x^{-1}+y+y^{-1}$ in the rotation algebra factor associat…

Last's periodic approximation conjecture. For any $\alpha\notin\mathbb{Q}$, $\mathcal{A}(\alpha)=\lim_{p/q\to\alpha}S_-(p/q)$, in the sense that

Let $(\beta,A_\epsilon^E)$ be an analytic almost periodic Jacobi cocycle, and suppose that its determinant is bounded away from zero. Almost reducibility conjecture for Jacobi cocy…

Let $(\beta,S_E)$ be the Schrödinger cocycle associated with an almost periodic Schrödinger operator having an analytic potential, and let the operator's spectrum be denoted by…

Let an almost-periodic operator have spectral decomposition into absolutely continuous, singular continuous, and pure point components, and let the phase vary. Simon’s phase-stabil…