71 problems

Rudnick–Sarnak conjecture. If $M$ is a compact Riemannian manifold of negative curvature, then every orthonormal basis of Laplacian eigenfunctions on $M$ should be quantum unique e…

Quantum unique ergodicity conjecture. It is not necessary to pass to a density-one subsequence: the entire sequence converges to the Liouville measure in the sense of semiclassical…

Let $\mathbb{H}:=\{x+iy:x\in\mathbb{R},y>0\}$, let $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, and let $\mathbb{X}=\Gamma\backslash\mathbb{H}$ with measure…

Let $(\Sigma,g)$ be a closed hyperbolic surface, and consider high-frequency limits of Laplace eigenfunctions on it. A semiclassical defect measure is a measure describing such a h…

Off-diagonal extension conjecture. The graph result of Brooks, Lindenstrauss, and Le Masson should extend to include off-diagonal elements, thereby enabling a proof of large-scale…

Let $X$ be a compact negatively curved manifold, let $\Delta_g$ be its Laplace operator, and let $\{\psi_{\lambda_j}\}_{j=1}^\infty$ be an orthonormal basis of eigenfunctions. For…

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 $(M,g)$ be a compact manifold of negative curvature without boundary, and let $(u_n)_{n\geq 1}$ be any orthogonal eigenbasis of the Laplacian. A semiclassical measure is a weak…

Let $\psi\colon \mathbb{R}^+ \to \mathbb{R}$ be a smooth, compactly supported function, and let $f(z)$ be an $L^2$-normalized holomorphic Hecke cusp form of weight $k$. Young's ver…

Nonsplit Bianchi quantum unique ergodicity conjecture. If $H\in C_b(\Gamma\backslash\mathbb{H})$ is fixed, then

Let $(M,g)$ be a hyperbolic surface, and let $u$ be a semiclassical measure arising as a weak-* limit of eigenfunctions on $S^M$. Colin de Verdière's conjecture. No semiclassical…

Let $S^*M$ be the unit cotangent bundle of a compact Riemannian manifold, decomposed into two flow-invariant sets of positive Liouville measure, with the geodesic flow ergodic on o…

Conjecture. Such sequences should not exist for systems associated with surfaces of constant negative curvature.

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$,…

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…

For a finite field $\mathbb{F}_{q}$, let $\mathcal{C}$ be the multiplicative group $\mathbb{F}_{q}^{*}$ or the group of norm-one elements in a quadratic extension, and let…

Let $M$ be a compact surface of negative curvature, let $-\Delta_g\psi_n=E_n\psi_n$ be Laplace–Beltrami eigenfunctions, let $\nu_g$ be the normalized Liouville measure on the unit…

Luo–Sarnak variance formula. The displayed asymptotic holds for the Hecke eigenfunction basis. The result is rigorous for the stated arithmetic surface and holomorphic Hecke eigenf…

For the quantization of an irrational skew translation on the two-torus, let $N$ denote the inverse Planck parameter and let the expectation values of observables in eigenstates co…

The Laplace–Beltrami operator on a surface of constant negative curvature has eigenfunctions, and one may consider its (Hecke) eigenfunctions and their semiclassical measures. Uniq…

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 $X$ be a compact Riemannian manifold, let $H:=\sqrt{-\Delta}$, and let $Q(f)$ be a pseudodifferential operator of order zero with principal symbol $f_P$ on $ST^*X$. Suppose the…

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues…

Böcherer–Sarnak–Schulze-Pillot conjecture. The basis $\{\psi_n\}$ of Hecke eigenfunctions of $L^2(\mathcal{O}^{\times}\backslash S^2)$ is quantum unique ergodic. The Jacquet–Langla…

Let $(\Sigma,g)$ be a negatively curved Riemannian surface and let $(L,\nabla)$ be a Hermitian line bundle with connection defining the magnetic Laplacian. In the high-energy regim…