Semiclassical measure for the solution of the dissipative Helmholtz equation

TL;DR

This paper analyzes the semiclassical measures of solutions to a dissipative Helmholtz equation with a source, revealing microlocal properties and the behavior of solutions in different regions.

Contribution

It introduces a microlocal analysis framework for dissipative Helmholtz equations with trapped trajectories passing through absorption regions.

Findings

Solutions decompose into Lagrangian distributions away from the source

Outgoing solutions are microlocally zero in incoming regions

The potential's trapping properties influence solution behavior

Abstract

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.