30 problems

Extremal-eigenstate convergence conjecture. If

Open-baker semiclassical-measure conjecture. All long-living semiclassical measures are supported on $\overline{\Gamma_+}$, and almost all long-living eigenstates are non-diffracti…

Limiting-distribution conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is that of the random variable

Feingold–Peres off-diagonal variance conjecture. For ergodic flow, as $E\to\infty$,

Let $\phi_n$ be Laplacian eigenfunctions on an ergodic billiard, and let a quantum limit mean a measure to which $|\phi_n|^2$ converges weakly. Quantum unique ergodicity conjecture…

Let $\Omega$ be a billiard domain and let $\hat{A}$ be multiplication by a test function $A$. Define the local diagonal variance $V_A(E)$ as the mean square of…

Let $\Omega$ be the billiard domain, let $\hat{A}$ be multiplication by a test function $A$, and define its spatial average by … For eigenfunctions $\phi_n$ of the Laplacian with e…

Let $H$ be a quantum Hamiltonian whose classical counterpart is integrable, and consider the point spectrum of $H$ (or its normalized energy differences). Berry's conjecture. The p…

Let $p$ be prime, let $\rho_{1/p}(A)$ be the quantized operator, and let ${\rm T}_A$ be its Hecke torus. For a character $\chi:{\rm T}_A\@ifnextchar^{\t@@}{\t@@^{}}\mathbb C^*$, le…

Let the Picard group be the arithmetic group underlying the spectrum of the Laplacian, and consider the sequence of its non-degenerate eigenvalues. Conjecture on density of non-deg…

The quantum mechanical system is desymmetrized with respect to all its unitary symmetries, eigenvalues are considered on the scale of the mean level spacing, and generic degeneraci…

Nodal universality conjecture. The variance has linear growth

Let $\Omega\subset\mathbb{R}^2$ be a chaotic billiard, so its billiard flow is ergodic and has positive topological entropy. Let $\{u_n\}$ be an orthonormal sequence of eigenfuncti…

Let $M$ be a compact complex hyperbolic quotient, and let $\mu$ be a semiclassical measure, meaning the semiclassical limit of a sequence of Laplacian eigenfunctions on $M$. Its su…

Let $(M,g)$ be a compact connected Riemannian manifold of negative sectional curvature. A weak limit $\tilde\mu$ is a weak- limit of the probability measures…

Fix $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on a bounded domain $D$, and let $(\mathbf{f}_n)$ be eigenfunctions normalised to have unit $L^2(\mu_\gamma)$ norm.…

Let a quantum system evolve under a time-independent Hamiltonian, and let $A$ be a subsystem. Entropy thermalization means that, after long-time evolution, the entropy of $A$ agree…

Let $\left\{\Gamma^{(\beta)}\right\}_{\beta\nearrow\infty}$ be any sequence of standard graphs, labeled by their first Betti numbers. For each $\Gamma^{(\beta)}$, choose arbitrary…

Let $\mathcal{H}^*=\{h_1,h_2,\ldots\}$ be an ordered set of integers. It is admissible if every finite truncation $\{h_1,\ldots,h_k\}$ is admissible, meaning that its reduction mod…

Let $H$ be the Hamiltonian, $Q$ a conserved charge, and define … For operators that can create states with arbitrarily large values of $Q$, consider the regularized out-of-time-ord…

Let $(M,g)$ be a Riemannian manifold, let $\varphi^t$ be its geodesic flow on $S^*M$, let $\mathcal M(\Delta_g)$ be the set of semiclassical measures of Laplace eigenfunctions, let…

Bohigas–Giannoni–Schmit conjecture. The eigenvalues are distributed like the eigenvalues of Hermitian random matrices, with the following limiting ensembles: without time-reversal…

Arithmetic quantum chaos conjecture. The distribution of the discrete eigenvalues approaches a Poisson distribution.

Let ${\mathcal L}_0$ be a Diophantine irrational lattice. Write $n_j$ for the distinct Laplacian eigenvalues, let $\delta_j=n_j-n_{j-1}>0$, let … and define the mean-normalized spa…

Bohigas–Giannoni–Schmit conjecture. If the classical problem is a sufficiently chaotic system, then the local statistics of the $E_n$ are generically those of EGO or EGU, according…