8 problems

Positive-integrality conjecture. For $\beta\in\{1,2\}$, the expansion coefficients satisfy

Let $g$ be a positive integer, let $\beta=1$ in the orthogonal case and $\beta=2$ in the unitary case, and let $\xi$ and $s$ be the variables used in the generating functions…

DMPK limiting-density conjecture. The density $\rho(x;\mathbb{M})$ is related to $\psi(x)$ by

Let ${\cal P}:=T^*{\bf T}$ be the classical phase space, with Hamiltonian $H(p,q):=\frac{1}{2}p^2+V(q)$ and Liouville measure $\lambda$. Let $\bar v:{\cal P}\to{\bf R}^d$ be the as…

Diagonal monitored-transport limit. If $L_N/N\to\tau$, then

Let $L(t)$ denote the limiting family of linear operators arising in the scaling limit where the quantum wavelength and scattering radius satisfy $h=r\to0$, with observables evolve…

Let $H_0$ be the Hamiltonian whose potential on the sites $n$ is the quasi-periodic function $v(n)=U\cos(2\pi\omega n)$, where $-2<U<2$ and $\omega$ is irrational. Assume that the…

Let $H$ be the coupled Hamiltonian, let $P^{d}(H)$ be the projection onto the subspace generated by the discrete eigenfunctions of $H$, and let ${\varrho}_{\alpha}(t)$ denote the t…