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

TL;DR

This paper introduces a new basis of vector functions called mixed vector spherical harmonics, simplifying the concentration problem of tangential vector fields within a spherical cap by reducing it to a scalar problem and providing a stable eigenfunction computation method.

Contribution

It develops a novel basis and a commuting differential operator approach to efficiently solve the vector field concentration problem on a spherical cap.

Findings

Reduced the vector concentration problem to a scalar problem

Constructed a differential operator that commutes with the concentration operator

Provided a stable method for computing eigenfunctions of the problem

Abstract

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.