On the completeness and the expansion theorem for eigenfunctions of the Sturm-Liouville-Rossby type

Abstract

Rossby waves in a closed ocean form an unfamiliar kind of eigenvalue problem. An investigation is made of the one-dimensional generalized version, which is called here the Sturm—Liouville—Rossby (SLR) problem. The eigenvalue and the eigenfunction of the SLR equation are calculated from those of the associated Sturm—Liouville (SL) equation and vice versa. The expansion theorem and the completeness theorem for the SLR eigenfunctions are proved to be valid in parallel to the SL problem. Two representations based on the SL eigenfunctions and on the SLR eigenfunctions are provided for Green’s function, which gives an example of a reproducing kernel.