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

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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Mathematics [math]

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Rellich-HAL.pdf (127.47 Ko)

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https://ensta-paris.hal.science/hal-01160242

Submitted on : Thursday, June 4, 2015-5:41:52 PM

Last modification on : Tuesday, August 6, 2024-10:56:09 AM

Long-term archiving on: Tuesday, September 15, 2015-11:12:01 AM

Dates and versions

hal-01160242 , version 1 (04-06-2015)

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HAL Id : hal-01160242 , version 1
DOI : 10.1016/j.crma.2015.10.015

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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, 2015, ⟨10.1016/j.crma.2015.10.015⟩. ⟨hal-01160242⟩

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