151 problems

In quantum chaos, consider a quantum system whose classical dynamics are sufficiently chaotic. Bohigas–Giannoni–Schmit conjecture. Its local spectral statistics should be described…

Berry's conjecture. The rescaled restriction of $\varphi_i$ to that wavelength-scale ball should behave like $\phi^{\mathbb{R}^n}_1$. This conjecture proposes a universal local mod…

Generic quantum integrable systems are quantum systems with integrable classical dynamics, and locally Poissonian level statistics means that their unfolded neighbouring energy lev…

Let a quantum system be chaotic, and consider the local features of its spectrum. Random matrix theory (RMT) is the statistical framework relating such spectral features to random…

Let $X$ be a typical hyperbolic surface, and consider the spectral statistics of its Laplacian. Bohigas–Giannoni–Schmidt conjecture. Because the geodesic flow is time-reversal symm…

Let $(X,g)$ be a smooth compact Riemannian surface of negative curvature, and let $\psi_\lambda\in L^2$ satisfy … Here $\\|\psi_\lambda\\|_{L^\infty}$ and…

Consider a thermal quantum system at temperature $T$, and let the out-of-time-order correlation (OTOC) quantify information scrambling. When the OTOC grows exponentially, write the…

Consider the Sachdev–Ye–Kitaev model of $N$ interacting Majorana fermions, with Majorana operators satisfying … Fast scrambling conjecture. The time required for an operator to "gr…

Let a quantum system arise as the quantization of a classically chaotic system, and consider its eigenvalues and their mean spacing. Random Matrix conjecture. Generically, the eige…

Let $A$ be an observable, let $V_A(\lambda)$ denote the variance of its diagonal matrix elements up to frequency $\lambda$, let $\omega(A)$ be its classical mean, and let…

Let a quantum chaotic system have highly excited eigenfunctions, and suppose the system is sufficiently disordered or quantum chaotic. Eigenstate Thermalization Hypothesis. The hig…

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…

Kurlberg–Rudnick's conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is the distribution of

The classical Hamiltonian dynamics is described by position $q$ and momentum $p$ satisfying … while the corresponding quantum Heisenberg operators $q_h$ and $p_h$ satisfy the analo…

Let a quantum system have a classically chaotic counterpart, and choose a basis in which to express one of its energy eigenstates. Berry's conjecture. The energy eigenstate should…

Gamma-distribution conjecture. The distributions $P_n(s)$ are well described by normalized gamma distributions

Let $\phi$ be a Maass cusp form on $\mathbb{X}$, let $\lambda_\phi$ be its Laplace eigenvalue, and let $N(\phi)$ denote the number of connected components of…

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

Consider the two-dimensional Gaussian random wave model and wave functions of chaotic billiards. Their nodal sets are the zero sets of the corresponding random waves or eigenfuncti…

Let $f$ be an observable and consider eigenstates $\psi_j$ in the small energy window $\hbar k_j\in[1-\varepsilon,1+\varepsilon]$. Their quantum averages are…

Energy-statistics correspondence conjecture. A regular classical system corresponds to Poisson energy-level statistics, while an irregular classical system corresponds to Wigner–Dy…

Quantum-chaos shadow conjecture. Quantum mechanics suppresses chaotic behavior so that only some “shadows” of chaos can remain.

Quantum-chaos suppression conjecture. Quantum mechanics suppresses chaotic behavior so that only “fingerprints” of chaos can remain.

Let $\Delta^{\rm Poisson}(L)$ and $\Delta^{\rm RMT}(L)$ denote the Poisson and random-matrix predictions for the spectral rigidity, and let $q$ be the irregularity fraction of the…

Extremal-eigenstate convergence conjecture. If