ar X iv : 0 80 9 . 08 26 v 1 [ m at h . FA ] 4 S ep 2 00 8 SELF - ADJOINT curl OPERATORS

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, D) equipped with a symplectic pairing arising from the ∧-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. 1. Introduction. The curl operator is pervasive in field models, in particular in electromagnetics, but hardly ever occurs in isolation. Most often we encounter a curl curl operator and its properties are starkly different from those of the curl alone. We devote the final section of this article to investigation of their relationship. The notable exception, starring a sovereign curl, is the question of stable force-free magnetic fields in plasma physics. They are solutions of the eigenvalue problem