11 problems

For a one-dimensional Schrödinger operator, consider the claim that the absolutely continuous spectrum is preserved in general under the relevant perturbations, with the perturbed…

Barry Simon's conjecture. The absolutely continuous spectrum of $H_V$ should fill the entire positive half-axis, that is, it should contain all of $(0,\infty)$.

Let $H=-\Delta+V(x)$ act on $\mathbb{R}^{d}$, where $V$ is a potential satisfying … Here $\sigma_{ac}(H)$ denotes the absolutely continuous spectrum of $H$, and $\mathbb{R}^{+}$ de…

Consider the half-line Schrödinger operator … acting on vector functions in $L^2(\mathbb{R}_+)$ with Neumann boundary condition $\psi'(0)=0$, where $v$ is a symmetric $n\times n$ m…

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…

For even $\kappa\geq6$, define $J_3(\kappa)=(\cos(2\pi/\kappa),\cos^2(\pi/\kappa))$, and let $\boldsymbol{\mu}_{\kappa}(D)$ denote the absolutely-continuous-spectrum set defined in…

Let $D+V$ be the discrete Molchanov–Vainberg Schrödinger operator on $^2(^2)$, and let the thresholds be the energies defined in the paper. Two-dimensional spectral-band conjecture…

Let $-\Delta$ be the discrete Laplacian on $\mathbb{Z}^3$, let $V$ be the potential under consideration, and let $\delta>0$ denote the disorder strength. Small-disorder absolutely…

Purely absolutely continuous spectrum conjecture. The spectrum of $H_0$ should be purely absolutely continuous as long as $b$ is non-constant.

Let $H_\lambda$ be the Anderson model on the Bethe lattice, with disorder strength $\lambda$, and let $\Sigma_\lambda$ be its almost-sure spectrum. The small-disorder pure ac spect…

Let $\mathcal G$ be a graph, let $\nu$ be a probability measure on $\mathbb R$, and let $q$ be a random potential defined through $\nu$ and satisfying the conditions stated above.…