Related papers: Remarks on quantum ergodicity

We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The…

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 deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^\infty (M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric…

We discuss problems that relate curvature and concentration properties of eigenfunctions and quasimodes on compact boundaryless Riemannian manifolds. These include new sharp $L^q$-estimates, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, of…

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 discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by…

This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential…

We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding…

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an…

This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de Verdi\`ere…

For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements…

This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a…

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the…

We state a theorem relating the ergodicity of the action of a given subgroup of the mapping class group of a surface on the character variety, to the asymptotic of its invariant subspaces through the Witten-Reshetikhin-Turaev…

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This…

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de…

This paper studies the uniformly asymptotic stability of nonautonomous systems on Riemannian manifolds. We establish corresponding Lyapunov-type theorems (Theorems 2.1 and 2.2), extending classical Euclidean results (e.g., [9, Theorems 4.9…

For systems evolving on a Riemannian manifold, we propose converse Lyapunov theorems for asymptotic and exponential stability. The novelty of the proposed approach is that is does not rely on local Euclidean coordinate, and is thus valid on…

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…