Eigenfunction expansions associated with the curl derivatives in cylindrical geometries: Completeness of Chandrasekhar–Kendall eigenfunctions

Eigenfunctions of the curl derivatives are used in many different fields of theoretical physics. For example, in plasma physics, an eigenfunction of curl represents a certain eigenstate of a plasma that has the force‐free property. The present theory proves that the spectral resolution of the curl operator gives a complete eigenfunction expansion of a solenoidal vector field with homogeneous boundary and flux conditions. Useful decomposition theories and a previous abstract result [Math. Z. 204, 235 (1990)] are concisely summarized. Explicit eigenfunctions have been studied in cylindrical geometries. Chandrasekhar–Kendall functions [Astrophys. J. 126, 457 (1957)], which are eigenfunctions of the curl derivatives in a cylindrical domain, give an orthogonal complete basis of the Hilbert space that is the orthogonal complement of the space of irrotational vector fields.