Spectral deviations for the damped wave equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction

Exponential decay for products of Fourier integral operators

This text contains an alternative presentation, and in certain cases an improvement, of the "hyperbolic dispersive estimate" that was proved by Anantharaman and Nonnenmacher and used to make progress towards the quantum unique ergodicity conjecture. The main statement is a sufficient condition to have exponential decay of the norm of a product of sub-unitary Fourier integral operators. The improved estimate will also be needed in future work of the author

Patterson-Sullivan distributions and quantum ergodicity

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}, \phi_{ir_j}>_{L^2(\X)}, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions \lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a (where the sum runs over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j \to \infty$. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.Comment: 54 pages, no figures. Added some reference

Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.Comment: We added the proof of the Entropic Uncertainty Principle. 45 pages, 2 EPS figure

Quantum Ergodicity on Graphs : from Spectral to Spatial Delocalization

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schr\"odinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schr\"odinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schr\"odinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply "quantum ergodicity", a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies in particular to graphs converging to the Anderson model on a regular tree, in the r\'egime of extended states studied by Klein and Aizenman-Warzel.Comment: To appear in the Annals of Mat