On the completeness of the spherical vector wave functions
In 1953, \textit{I. N. Vekua} [Dokl. Akad. Nauk SSSR 90, 715-718 (1953; Zbl 0051.049)] developed methods for studying \(L^ 2\)-completeness properties of sets of spherical wave functions on a Lyapunov surface \(S\) bounding a finite domain \(D_ i\). The authors review previous work on scalar and vector problems, and extend Vekua's procedure to the case of spherical vector wave functions (SVWF) that satisfy the vector Helmholtz equation \((\nabla^ 2+k^ 2)F=0\). They show that the set of regular SVWF is complete on \(S\) if \(k^ 2\) is not an eigenvalue of the interior Dirichlet problem. If \(k^ 2\) is such an eigenvalue, completeness is restored by appending the normal derivatives on \(S\) of the vector eigenfunctions. When \(k^ 2\) is not a Dirichlet eigenvalue, it is shown that a vector function that is continuous on \(D_ i\cup S\) can be approximated uniformly on closed subsets of \(D_ i\) and in the mean on \(S\) by a sequence of linear combinations of regular SVWF. Corresponding results that use the set of radiating SVWF are obtained for exterior problems, in which case no eigenvalue restriction arises. Finally, the extension is made to problems in a finite domain bounded internally and externally by closed Lyapunov surfaces.
scientific article; zbMATH DE number 3996201
scientific article