Spectral theory of pseudo-differential operators of degree 0 and application to forced linear waves

We extend the results of our paper "Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0" written with Laure Saint-Raymond to the case of forced linear wave equations in any dimension. We prove that, in dimension 2,if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefrontsets of the limit Schwartz distribution using radial propagation estimates

The spectrum of the Poincar{\'e} operator in an ellipsoid

We reprove the fact, due to Backus, that the Poincar{\'e} operator in ellipsoids admits a pure point spectrum with polynomial eigenfunctions.We then show that the eigenvalues of the Poincar{\'e} operator restricted to polynomial vector fields of fixed degree admitsa limit repartition given by a probability measure that we construct explicitely. For that, we use Fourier integral operators and ideas comingfrom Alan Weinstein and the first author in the seventies. The starting observation is that the orthogonal polynomials in ellipsoids satisfy a PDE

On essential-selfadjointness of differential operators on closed manifolds

The goal of this note is to present some arguments leading to the conjecture that a formally self-adjoint differential operator on a closed manifold is essentially self-adjoint if and only if the Hamiltonian flow of its symbol is complete. This holds for differential operators of degree two on the circle, for differential operators of degree one on any closed manifold and for generic Lorentzian Laplacians on surfaces

Spectral theory of pseudo-differential operators of degree 0 and application to forced linear waves

International audienceWe extend the results of our paper "Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0"written with Laure Saint-Raymond to the case of forced linear wave equations in any dimension. We prove that, in dimension 2,if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefrontsets of the limit Schwartz distribution using radial propagation estimates

Large time asymptotics of the wave fronts length II:surfaces with integrable Hamiltonians

In a previous work, David Vicente gave a formula showing that the wave front issued of a point of the unit disk hasa large time linear asymptotics. In the present paper, we extend the result to intgrable 2D Hamiltoniansystems

Tunneling on graphs: an approach ``à la Helffer-Sjöstrand''

we revisit a paper of Li-Gabor-Yau on quantum tunneling on graphs using the approach of Helffer-Sjöstrand for the study of tunneling for Schrödinger operators on manifolds

Tunneling on graphs: an approach ``à la Helffer-Sjöstrand''

we revisit a paper of Li-Gabor-Yau on quantum tunneling on graphs using the approach of Helffer-Sjöstrand for the study of tunneling for Schrödinger operators on manifolds

Spectral theory of pseudo-differential operators of degree 0 and application to forced linear waves

International audienceWe extend the results of our paper "Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0"written with Laure Saint-Raymond to the case of forced linear wave equations in any dimension. We prove that, in dimension 2,if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefrontsets of the limit Schwartz distribution using radial propagation estimates

Periodic geodesics for contact sub-Riemannian 3D manifolds

The goal of this paper is to study periodic geodesics for sub-Riemannian metrics on a contact 3D-manifold.We develop two rather independent subjects:1) The existence of closed geodesics spiraling around periodic Reeb orbits for a generic metric.2) The precise study of the periodic geodesics for a right invariant metric on a quotient of SL2(R

Topological Resonances on Quantum Graphs

International audienceIn this paper, we try to put the results of Smilansky and al. on "Topological resonances" on a mathematical basis.A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exists compactly supported eigenfunctions. We give several estimateof the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case oftrees is also discussed