Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem

Anne-Sophie Bonnet-Ben Dhia (1) , Lucas Chesnel (2) , Vincent Pagneux (3)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LAUM - Laboratoire d'Acoustique de l'Université du Mans

We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of PT-symmetric problems.

Mots clés: backscattering; trapped modes; transmission resonances; waveguides; complex scaling; PT symmetry

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Physique mathématique [math-ph]

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Vincent Pagneux. Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, 2018. ⟨hal-01692297v2⟩