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

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.