Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution

We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions $F_{lm}$ of Sheppard and T"or"ok and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Gr"unbaum and his colleagues, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.