2025/04/24 by Humaikani, Sarah Al, Dhia, Anne-Sophie Bonnet-Ben, Fliss, Sonia +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2504.17345
We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than π/2.