Related papers: Quantum Ergodicity for Point Scatterers on Arithme…
We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori $\mathbb{R}^d/2 \pi\mathbb{Z}^d$ in dimensions $d=2,3$. Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that…
We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j},…
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. These systems cannot be obtained as the quantization of a classical Hamiltonian, as the…
We study the small scale distribution of the $L^2$ mass of eigenfunctions of the Laplacian on the flat torus $\mathbb T^d$. Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose $L^2$ mass…
We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely…
We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of…
For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one…
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…
This survey article deals with a delta potential - also known as a point scatterer - on flat 2D and 3D tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so- called weak and…
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)…
We prove quantitative equidistribution properties for orthonormal bases of eigenfunctions of the Laplacian on the rational $d$-torus. We show that the rate of equidistribution of such eigenfunctions is of polynomial decay. We also prove…
Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schr\"odinger operator. Under a geometric condition on the magnetic field, we show…
We apply the techniques of our previous paper to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of…
For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we…
We define a random model for the moments of the new eigenfunctions of a point scat-terer on a 2-dimensional rectangular flat torus. In the deterministic setting,Seba conjectured these moments to be asymptotically Gaussian, in the…
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…
We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum…
We study the small scale distribution of the $L^2$-mass of eigenfunctions of the Laplacian on the the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick showed the existence of a density one…
We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any probability measure…