14 problems

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…

Let $H_1$ and $H_2$ be the operators described in the preceding theorem, and let $4e2_n$ denote the singular values of the relevant difference of semigroups, with spatial dimension…

Continuity conjecture. The measure $\mu-\mu^{ND}$ is continuous, i.e. it does not charge any point.

Consider the universal Schwarzian sector of three-dimensional gravity and an analogous sum over three-dimensional geometries. Airy-density conjecture. The sum over $3$-geometries m…

Let a quantum field theory have a finite group global symmetry $\mathrm{G}$ and be defined on a compact spatial manifold. For each irreducible representation (irrep) $\alpha$ of…

Let $\mathcal{T}$ be an arbitrary universal cover, and let $A_{\mathcal{T}}$ be the associated Jacobi operator. Suppose the cumulative distribution function of the density of state…

Fourier-transform decay conjecture. The condition $$ in [H5] should hold for $f706$ and $dgeq 3$.

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 $[?]$ be a random Schrödinger model in arbitrary dimension, with single-site distribution $[?]$ and characteristic function $\widehat{\mu}(t)$. The conjecture concerns relative…

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…

Monotonicity conjecture. The function $d_\lambda$ is monotone in $\lambda$.

Let $d\geq 4$ and let $\mu_{L,E}^0$ be the measures associated with the finite-volume model at energy $E\in(-2d,2d)$. Let $n_d$ denote the density of states of the $d$-dimensional…

Band-center singularity conjecture. As $E\to 0$,

The asymptotic expansion conjecture. For large $\lambda$, there are coefficients $e_j$ such that