Julien Royer

Limiting absorption principle for the dissipative Helmholtz equation

Adapting Mourres commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the uniform resolvent estimate for the high-frequency Helmholtz equation when trapped classical trajectories meet the region where the absorption coefficient is non-zero. We also give the resolvent estimate in Besov spaces.

Local Energy Decay for the Damped Wave Equation

Abstract We prove local energy decay for the damped wave equation on R d . The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding “resolvent”, both for low and high frequencies. These estimates are given by an improved dissipative version of Mourres commutators method.

Semiclassical measure for the solution of the dissipative Helmholtz equation

We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.

Local energy decay and smoothing effect for the damped Schrödinger equation

We prove the local energy decay and the smoothing effect for the damped Schrodinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.

Local decay for the damped wave equation in the energy space

We improve a previous result about the local energy decay for the damped wave equation on R^d. The problem is governed by a Laplacian associated with a long range perturbation of the flat metric and a short range absorption index. Our purpose is to recover the decay O(t^{--d+

Exponential Decay for the Schrödinger Equation on a Dissipative Waveguide

\epsilon

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

}) in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

On the Cauchy Problem and the Black Solitons of a Singularly Perturbed Gross--Pitaevskii Equation

We prove exponential decay for the solution of the Schrödinger equation on a dissipative waveguide. The absorption is effective everywhere on the boundary, but the geometric control condition is not satisfied. The proof relies on separation of variables and the Riesz basis property for the eigenfunctions of the transverse operator. The case where the absorption index takes negative values is also discussed.

Reduction of dimension as a consequence of norm-resolvent convergence and applications

We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.

Semiclassical measure for the solution of the Helmholtz equation with an unbounded source

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.