37 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 $H_{\rm Anderson}=-\lambda\Delta+V$ be the Anderson model on the lattice $\mathbb{Z}^d$ for $d\ge 3$, where $\Delta$ is the discrete Laplacian, $V$ is a random potential with i…

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…

A disordered potential is a potential introducing spatial randomness into a lattice Schrödinger operator, and localization means that the operator has purely point spectrum with ei…

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 $W$ denote the bandwidth of an $N\times N$ random band matrix, interpolating between a one-dimensional random Schrödinger operator when $W\sim 1$ and a Wigner matrix when…

Let $u_1({\mathbf r})$ be the first-order localization landscape function, and let $v_2({\mathbf r})$ and $\phi_2({\mathbf r})$ satisfy … … Define the second-order landscape functi…

Single-parameter-scaling conjecture. Distribution functions of physical observables, like the conductance, are effectively controlled by a single parameter.

Let $H=H_\beta=-\beta\Delta+V$ be the Anderson Hamiltonian on a discrete torus $\Lambda$ approximating $\mathbb{Z}^d$, and let … In the same setting, the more precise quantum diffu…

Anderson delocalisation conjecture. For the Anderson model, this occurs in $d\geq3$ for $E$ inside the spectrum of $-\beta\Delta$ when $\beta$ is sufficiently large. This is a long…

The Anderson Hamiltonian $d539^{(E)}$ has eigenvectors whose rescaled and rotated profiles are described by the limiting critical stochastic operator $d4c2\mathcal{S}_\tau$. More p…

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…

Let the almost Mathieu potential be $v(\theta)=\cos (2\pi \theta)$. Consider the corresponding two-particle quasiperiodic model with an interaction potential. FIKS conjecture. Ther…

The model describes disordered systems with a non-compact hyperbolic symmetry, and extended states are associated with spontaneous breaking of that symmetry. In dimension…

Let $d=2$ and $v(x)=\cos x$, so that the Schrödinger equation becomes the Almost Mathieu equation … Here $L(E)$ denotes the Lyapunov exponent, and $M_{(x,y),\omega,\lambda v}$ is t…

Let $H$ be the hierarchical Anderson model with parameter $c>0$, and let $d_s$ denote its spectral dimension. Absence-of-delocalization conjecture. There is no delocalization trans…

Consider the lattice Bernoulli–Anderson Hamiltonian in dimension $d>1$, and let the density of states (DoS) denote the density associated with its integrated density of states. The…

Let the one-dimensional Bernoulli–Anderson model be given in a parameter zone where the integrated density of states (IDS) is considered. An IDS is Hölder continuous of order…

Let ${\mathbf s}$ be a sequence and let $H_\lambda={\mathcal A}_{\mathbf s}+\lambda {\mathcal V}$ be the Anderson model on the antitree ${\mathbb A}_{\mathbf s}$, with disorder par…

Let $[a,b]$ be a compact interval and let ${\mathbf s}$ be any sequence. Let $H_\lambda$ be the Anderson model on the antitree ${\mathbb A}_{\mathbf s}$ with disorder parameter…

Let $\mathcal{H}_{V}$ be the finite-volume Anderson Hamiltonian on a finite set $V\subset\mathbb{Z}^{\nu}$, and let $V\uparrow\mathbb{Z}^{\nu}$. Consider the eigenvalues of…

Consider the Anderson Hamiltonian … on the Hilbert space $\ell^2({\mathbb Z}^d)$, where $\Delta$ is the lattice Laplacian and $V(x)$ is a real-valued random potential whose values…

Anderson localization conjecture. Anderson localization holds for random numbers $F433$ with probability one, with a non-random localization length.

Let $H_{\lambda}$ be the Anderson Hamiltonian on a Bethe lattice, with bounded random potential having a sufficiently regular probability distribution, and let $L_{\lambda}(E)$ den…

Let $H_{\lambda}$ be the Anderson Hamiltonian on a Bethe lattice of branching number $K$, with disorder parameter $\lambda$. The free operator has an $l^{2}$ spectral threshold at…