Many interacting particles in solution. III. Spectral analysis of the associated Neumann--Poincar\'e-type operators

Abstract

The interaction of particles in an electrolytic medium can be calculated by solving the Poisson equation inside the solutes and the linearized Poisson--Boltzmann equation in the solvent, with suitable boundary conditions at the interfaces. Analytical approaches often expand the potentials in spherical harmonics, relating interior and exterior coefficients and eliminating some coefficients in favor of others, but a rigorous spectral analysis of the corresponding formulations is still lacking. Here, we introduce pertinent composite many-body Neumann--Poincar\'e-type operators and prove that they are compact with spectral radii strictly less than one. These results provide the foundation for systematic screening-ranged expansions, in powers of the Debye screening parameters, of electrostatic potentials, interaction energies, and forces, and establish the analytical framework for the accompanying works arXiv:2512.09421, arXiv:2512.08407, arXiv:2512.08682.