11 problems

Let a continuous random Schrödinger operator act on $L^2(\mathbb{R})$, and let $E_0$ be the bottom of its spectrum. Its integrated density of states is denoted by $N(E)$. Lifshitz'…

Borderline Lifshits-tail conjecture. As $E\downarrow0$,

Let the integrated density of states (IDS) be the spectral distribution measure associated with a Schrödinger operator having a random potential. Lifshitz's conjecture. The IDS is…

Sabot–Tarrès–Zeng conjecture. The asymptotic behavior of the IDS satisfies

Sabot–Tarrès–Zeng conjecture. The asymptotic behavior of the IDS of $H_\beta$ does not exhibit Lifshitz tails.

Let $p$ be the Bernoulli parameter, let $\beta^{-1}(n)$ denote the decreasing sequence appearing in the model, and let $I_{p,\zeta}(x)$ be the integrated density of states at disor…

Spencer's conjecture. The integrated density of states for the full Hamiltonian should be at least $1/2$-Hölder continuous.

Infinite-flatness conjecture. When the random variables do not have this Bernoulli distribution, the integrated density of states should be infinitely flat at $\inf\Sigma^D$.

Consider the one-dimensional Bernoulli displacement model with coupling parameter $\lambda$, integrated density of states $N$, and almost-sure spectrum $\Sigma_{\lambda}$. For…

Spectral-band conjecture. For every $\lambda\neq 0$,

Let $H=-\Delta+b$ act in $L_2(\mathbb R^d)$, where $b=b(\mathbf x)$ is real, smooth, and periodic, and let $N(\lambda)$ denote the integrated density of states of $H$. For the unpe…