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Self-adjoint curl operators

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 $\partial D$. 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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Submitted on : Friday, December 13, 2019 - 10:53:49 PM
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Ralf Hiptmair, P. Robert Kotiuga, Sébastien Tordeux. Self-adjoint curl operators. Annali di Matematica Pura ed Applicata, Springer Verlag, 2012, 191 (3), pp.431-457. ⟨10.1007/s10231-011-0189-y⟩. ⟨inria-00527733⟩

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