Uniform resolvent estimates for a non-dissipative Helmholtz equation

Julien Royer (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: Non-selfadjoint operators; resolvent estimates; limiting absorption principle; Helmholtz equation; semiclassical measures

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Julien Royer. Uniform resolvent estimates for a non-dissipative Helmholtz equation. Bulletin de la société mathématique de France, 2014, 142 (4), pp.591--633. ⟨hal-00578417⟩