UNIFORM RESOLVENT ESTIMATES FOR A NON-DISSIPATIVE HELMHOLTZ EQUATION

摘要

We study the high frequency limit for a non-selfadjoint Helmholtz equation. This equation models the propagation of the electromagnetic field of a laser in an inhomogeneus material medium with non-constant absorption index. In this paper the absorption index can take negative values and we only use a damping condition on the classical limit of the problem. In this setting we first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies this damping assumption. Then we generalize 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.