Related papers: Quantum ergodic restriction theorems, I: interior …

We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate…

We prove that if $(M, g)$ is a compact Riemannian manifold with ergodic geodesic flow, and if $H \subset M$ is a smooth hypersurface satisfying a generic asymmetry condition with respect to the geodesic flow, then restrictions $\phi_j |_H$…

We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces $\Sigma \subset X,$ where $X$ is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain $h$ independent upper…

Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple…

We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of…

For sequences of quantum ergodic eigenfunctions, we define the quantum flux norm associated to a codimension $1$ submanifold $\Sigma$ of a non-degenerate energy surface. We prove restrictions of eigenfunctions to $\Sigma$, realized using…

For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric…

This paper is a proceedings version of \cite{CHT-I}, in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL). We consider a sub-Riemannian (sR)…

Suppose that Omega is a bounded, piecewise smooth Euclidean domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Omega with various boundary conditions are quantum ergodic if the classical billiard…

We study the ergodic properties of eigenfunctions of Schr\"odinger operators on a closed connected Riemannian manifold $M$ in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an…

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric…

This article is about two types of restrictions of eigenfunctions $\phi_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $\gamma_H \phi_j$ in eigenfunctions $e_k$ of…

We prove a quantum ergodicity theorem for sequences of closed hyperbolic surfaces converging to the Poincar\'e disc in the Benjamini-Schramm sense. Assuming a uniform lower bound on the injectivity radius and a spectral gap, we establish…

Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property…

We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space…

This paper is dedicated to the proof of a Quantum Ergodicity (QE) theorem for the eigenfunctions of subLaplacians on contact metric manifolds, under the assumption that the Reeb flow is ergodic. To do so, we rely on a semiclassical…

The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility,…

This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results…

Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero…

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z)$, which we take to be ergodic. Under some natural…