arXiv subjects

Julien Royer

Publications and source records attributed to Julien Royer.

31 records · Page 2Linked to original sources

arXiv2016

On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation

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.

math.AP↗

arXiv2016

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

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.

math.SP↗

arXiv2016

Local Energy Decay and Diffusive Phenomenon in a Dissipative Wave Guide

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is based on resolvent estimates. Since the eigenvectors for the transverse operator do not form a Riesz basis, the spectral analysis does not trivially reduce to separate analyses on compact and Euclidean domains.

math-ph↗

arXiv2015

Exponential decay for the Schr{ö}dinger equation on a dissipative waveguide

We prove exponential decay for the solution of the Schr{ö}dinger 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.

math-ph↗

arXiv2015

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+$ε$}) 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.

math-ph↗

arXiv2015

Local energy decay and smoothing effect for the damped Schr{ö}dinger equation

We prove the local energy decay and the smoothing effect for the damped Schr{ö}dinger 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.

math-ph↗

arXiv2014

Mourre's method for a dissipative form perturbation

We prove uniform resolvent estimates for an abstract operator given by a dissipative perturbation of a self-adjoint operator in the sense of forms. For this we adapt the commutators method of Mourre. We also obtain the limiting absorption principle and uniform estimates for the derivatives of the resolvent. This abstract work is motivated by the Schr{ö}dinger and wave equations on a wave guide with dissipation at the boundary.

math.SP↗

arXiv2014

Sharp low frequency resolvent estimates on asymptotically conical manifolds

On a class of asymptotically conical manifolds, we prove two types of low frequency estimates for the resolvent of the Laplace-Beltrami operator. The first result is a uniform $ L^2 \rightarrow L^2 $ bound for $ \langle r \rangle^{-1} (- Δ_G - z)^{-1} \langle r \rangle^{-1} $ when $ \mbox{Re}(z) $ is small, with the optimal weight $ \langle r \rangle^{-1} $. The second one is about powers of the resolvent. For any integer $N$, we prove uniform $ L^2 \rightarrow L^2 $ bounds for $ \langle εr \rangle^{-N} (-ε^{-2} Δ_G - Z)^{-N} \langle εr \rangle^{-N} $ when $ \mbox{Re}(Z) $ belongs to a compact subset of $ (0,+\infty) $ and $ 0 < ε\ll 1 $. These results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity.

math.AP↗

arXiv2013

Local Energy Decay for the Damped Wave Equation

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 Mourre's commutators method.

math-ph↗

arXiv2013

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

We study the high frequency limit for the dissipative Helmholtz equation when the source term concentrates on a submanifold of R^n. We prove that the solution has a unique semi-classical measure, which is precisely described in terms of the classical properties of the problem. This result is already known when the micro-support of the source is bounded, we now consider the general case.

math-ph↗

arXiv2011

Uniform resolvent estimates for a non-dissipative Helmholtz equation

We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.

math-ph↗

arXiv2009

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.

math-ph↗

arXiv2009

Limiting absorption principle for the dissipative Helmholtz equation

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.

math.AP↗