47 problems

Let $H_\omega=-\Delta+V_\omega$ be the Anderson model on the $d$-dimensional Sierpinski simplex graph $(\mathbb{V},\mathcal E)$ for $d\ge 2$. Anderson localization conjecture. For…

Let $H$ be the Anderson model in dimension $d=2$. Two-dimensional Anderson localization conjecture. The spectrum of $H$ is pure point with dynamically localized eigenfunctions. Thi…

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'…

A random displacement model is a Schrödinger operator in which single-site potentials are randomly displaced; in dimension one, its spectral minimum is the infimum of the spectrum…

Let $H(V)$ be the ergodic random Schrödinger operator considered in the paper, with vector potential $a=0$, and let $N(E)$ denote its integrated density of states. The density of s…

For a random operator with disorder, consider the energy levels of finite-volume restrictions on the scale of their typical energy spacing. Write “pp” for pure-point spectral regim…

Averaged local single-site SSF conjecture. In this setting, the averaged SSF should belong to

Consider the Anderson Hamiltonian … on $\ell^2(\mathbb Z^d)$, where $\lambda>0$ is the coupling constant and $V_{\omega}$ is an i.i.d. random potential. For sufficiently large coup…

A discrete one-dimensional random word model is determined by a word space whose richness controls the exceptional energies excluded from the localization results. Absence-of-excep…

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

Let $N(\lambda)$ be the eigenvalue counting function of the Schrödinger operator considered in the paper, and define … Here $\kappa$ is the parameter appearing in the model. The ei…

Let $H$ be the Schrödinger operator considered in the paper, let $N(\lambda)$ be its eigenvalue counting function, and let $R_{\mathrm h}(\lambda)$ denote the half-line hyperunifor…

Let the Anderson model be a Schrödinger operator on $\mathbb{Z}^d$ whose random potential is uniformly distributed over an interval. Anderson's conjecture. At sufficiently strong d…

Let an eigenvector of an operator on a lattice have sufficiently fast exponential decay. Landis conjecture. Such exponential decay implies that the eigenvector is compactly support…

Let $H=-\Delta+\lambda V$ be the discrete random Schrödinger operator on $\mathbb{Z}^{d}$, where $\Delta$ is the graph Laplacian, $V$ is a random potential with independent identic…

Consider the random block Schrödinger operators of the source, with coupling parameter $\lambda$ and model-dependent scale $\Lambda_{\Psi}$. Critical-threshold conjecture. The crit…

Consider the random block Schrödinger operators studied in the source, with coupling parameter $\lambda$ and model-dependent scale $\Lambda_{\Psi}$. Let $\lambda_c$ denote the crit…

Let $H=-\Delta+\delta V$ be the $d$-dimensional Anderson model on $\ell^2(\mathbb{Z}^d)$, where $\Delta$ is the lattice Laplacian, $V$ is a random potential with independent identi…

Gaussian fluctuations conjecture. As $\lambda\to\infty$,

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.

A random Schrödinger operator is an operator of the form $H=H_0+V_\omega$, where $H_0$ is a deterministic Schrödinger operator and $V_\omega$ is a random potential. In the localiza…

Anderson's conjecture. Every initial condition can be almost surely decomposed into two parts: a low-energy part that remains dynamically localized and a bulk-energy part that prop…

White-noise semigroup variance conjecture. One has

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…