Characterization of the Eigenvalues and Eigenfunctions of the Helmholtz Newtonian operator N^k

TL;DR

This paper analyzes the eigenvalues and eigenfunctions of the Helmholtz Newtonian operator, providing explicit calculations for a 3D ball and exploring their asymptotic behavior with numerical illustrations.

Contribution

It establishes the eigensystem equivalence for the Newtonian potential operator and explicitly computes eigenvalues and eigenfunctions in a 3D ball, including asymptotic analysis.

Findings

Eigenvalues and eigenfunctions explicitly computed for a 3D ball

Eigenvalues' asymptotic behavior demonstrated

Numerical simulations illustrate eigenfunction behavior

Abstract

The Newtonian potential operator for the Helmholtz equation, which is represented by the volume integral with fundamental solution as kernel function, is of great importance for direct and inverse scattering of acoustic waves. In this paper, the eigensystem for the Newtonian potential operator is firstly shown to be equivalent to that for the Helmholtz equation with nonlocal boundary condition for a bounded and simply connected Lipschitz-regular domain. Then, we compute explicitly the eigenvalues and eigenfunctions of the Newtonian potential operator when it is defined in a 3-dimensional ball. Furthermore, the eigenvalues' asymptotic behavior is demonstrated. To illustrate the behavior of certain eigenfunctions, some numerical simulations are included.