Positivity of a weakly singular operator and approximation of wave scattering from the sphere

Abstract

We investigate properties of a family of integral operators $\mathcal{ℬ}$ with a weakly singular compactly supported zonal kernel function on the surface $S$ of the unit 3D sphere. The support is over a spherical cap of height $h\in (0,2]$. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where $h$ is directly related to the time step size.

We show that the Legendre polynomials of degree $\ell \ge 0$ satisfy for all $h\in (0,2]$ and, using spherical harmonics and the Funk–Hecke formula for the eigenvalues of $\mathcal{ℬ}$, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of $\mathcal{ℬ}$ and on ${\int }_S\hspace{0.17em}u(\mathit{x})(\mathcal{ℬ}u)(\mathit{x})d\mathit{x}$. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of $\mathcal{ℬ}$ is a necessary condition for these approximation schemes to be well defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ${\int }_0^r\hspace{0.17em}{J}_0(z)dz>0$ for all $r>0$.