A Rellich type theorem for the Helmholtz equation in a conical domain

Anne-Sophie Bonnet-Ben Dhia 1 Sonia Fliss 1 Christophe Hazard 1 Antoine Tonnoir 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We prove that there cannot exist square-integrable nonzero solutions to the Helmholtz equation in an axisymmetric conical domain whose vertex angle is greater than π. This implies in particular the absence of eigenvalues embedded in the essential spectrum of a large class of partial differential operators which coincide with the Laplacian in the conical domain.
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Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Christophe Hazard, Antoine Tonnoir. A Rellich type theorem for the Helmholtz equation in a conical domain. Comptes Rendus Mathématique, Elsevier Masson, 2015, ⟨10.1016/j.crma.2015.10.015⟩. ⟨hal-01160242⟩

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