Self-adjoint curl operators

2008/09/04 by Ralf Hiptmair, R. Hiptmair, Hiptmair, R. +6
Computer Science · Engineering · Mathematics · #46N20 #47F05 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math.FA #msc:46N20 #msc:47F05

paper · pdf · doi:10.48550/arxiv.0809.0826

30 pages, no figures

arxiv created 2008/09/04 · openalex publication_date 2008/09/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic pairing arising from the \wedge-product of 1-forms on ∂ D. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.

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