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

TL;DR

This paper investigates the high-frequency behavior of solutions to the dissipative Helmholtz equation with unbounded sources, establishing the uniqueness and precise description of the semiclassical measure in this more general setting.

Contribution

It extends previous results by characterizing the semiclassical measure for sources concentrated on submanifolds without the boundedness restriction.

Findings

Unique semiclassical measure for the solution

Explicit description in terms of classical properties

Generalization to unbounded source cases

Abstract

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.