16 problems

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…

Let $H(p,q):=\frac{1}{2}p^2+V(q)$ be the classical Hamiltonian on ${\cal P}:=T^*{\bf T}$, where ${\bf T}:={\bf R}^d/{\cal L}$ and $V\in C^\infty({\bf R}^d,{\bf R})$ is ${\cal L}$-p…

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

Quantum square-root conjecture. The appearance of a square root to preserve the triangle inequality could be a specifically quantum feature of optimal transport costs.

Monotonicity conjecture. For dimensions $N=3$ and $N=4$, the optimal quantum transport cost is monotonous under every CPTP map:

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

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…

In the periodic quantum transport setting, let the linear Boltzmann equation describe the limiting particle density in the Boltzmann–Grad and high-frequency scaling limits. Higher-…

The semiclassical or hydrodynamic approach is used to study inhomogeneous quenches and physical observables at large times and distances. Semiclassical approach conjecture. Even th…

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…

Consider a one-dimensional ergodic Schrödinger operator modeling a sample attached to two electronic reservoirs, with transport described by the Landauer–Büttiker and Thouless cond…

Half-filling Drude-weight conjecture. After including all local conserved charges, so that the Mazur bound is expected to saturate, the Drude weight at half-filling should vanish.

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…

Let $H_\lambda$ be the Anderson Hamiltonian on the Bethe lattice, with disorder strength $\lambda>0$. If transport occurs and its transport index $\alpha\in(0,1]$ is defined by the…

Property RST. For every potential $v$, the half-line Schrödinger operator exhibits regular spectral transport: for any choice of the reservoirs,

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…