Self-adjoint curl operators

TL;DR

This paper characterizes boundary conditions for the curl operator on 3D domains to be self-adjoint, using symplectic geometry and cohomology, and discusses their spectral properties and relation to curl curl operators.

Contribution

It generalizes previous results by characterizing all self-adjoint extensions of the curl operator via Lagrangian subspaces, incorporating domain topology and cohomology.

Findings

All self-adjoint extensions have discrete spectra.

Boundary conditions are characterized by Lagrangian subspaces of the trace space.

The relationship between curl and curl curl operators is clarified.

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.