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Many interacting particles in solution. III. Spectral analysis of the associated Neumann--Poincaré-type operators

Sergii V. Siryk, Walter Rocchia

TL;DR

This paper develops a rigorous spectral analysis for many-body Neumann–Poincaré–type operators arising from a coupled Poisson–Poisson–Boltzmann problem for multiple dielectric spheres in an electrolyte. By reformulating the problem via boundary-integral equations and representing the external potentials in spherical-harmonic coefficients, the authors prove that the associated NP-type operators are compact with $r<1$, enabling absolutely convergent Neumann-series solutions and screening-ranged expansions of potentials, energies, and forces. The results bridge continuous BIE formulations with infinite-dimensional coefficient spaces, establishing existence, uniqueness, and stability while providing a solid mathematical basis for efficient, distance-decaying expansions in complex colloidal/biomolecular systems. The numerical experiments corroborate the theoretical bounds, showing spectral radii stay below unity and decay as interparticle distance grows, thus supporting practical applicability to biomolecular electrostatics and materials design in electrolytes.

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--Poincaré-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.

Many interacting particles in solution. III. Spectral analysis of the associated Neumann--Poincaré-type operators

TL;DR

This paper develops a rigorous spectral analysis for many-body Neumann–Poincaré–type operators arising from a coupled Poisson–Poisson–Boltzmann problem for multiple dielectric spheres in an electrolyte. By reformulating the problem via boundary-integral equations and representing the external potentials in spherical-harmonic coefficients, the authors prove that the associated NP-type operators are compact with , enabling absolutely convergent Neumann-series solutions and screening-ranged expansions of potentials, energies, and forces. The results bridge continuous BIE formulations with infinite-dimensional coefficient spaces, establishing existence, uniqueness, and stability while providing a solid mathematical basis for efficient, distance-decaying expansions in complex colloidal/biomolecular systems. The numerical experiments corroborate the theoretical bounds, showing spectral radii stay below unity and decay as interparticle distance grows, thus supporting practical applicability to biomolecular electrostatics and materials design in electrolytes.

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--Poincaré-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.

Paper Structure

This paper contains 19 sections, 9 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Boundary value problems and equivalent Boundary-Integral-Equations formulations
  3. Problem statement
  4. Layered potential operators and general boundary-integral-equations (BIE) formulations
  5. Representation in the spherical Fourier coefficients space
  6. Spectral analysis of the associated NP-type operators
  7. Numerical examples
  8. Proofs of the propositions
  9. Mathematical preliminaries and a brief summary of facts on compact and Fredholm operators used in the proofs
  10. Proof of Proposition \ref{['proposition_matrix_repr']}
  11. Proof of Proposition \ref{['proposition_K_compact']}
  12. Proof of Proposition \ref{['proposition_spectral_radius']}
  13. Proofs of Corollaries \ref{['corollary_Neumann_covergence']} and \ref{['corollary_solution_exist_unique']}
  14. Proof of Corollary \ref{['corollary_solutions_real']}
  15. On the convergence of series \ref{['scr_ranged_expansion_G']}-\ref{['energy_expansion_components_abs_gen']}
  16. ...and 4 more sections

Key Result

Proposition 1

System global_lin_sys1 is an $\pmb l^2$-representation of operator identity main_eq_G_op_Sobolev in the basis $\bigoplus_{i=1}^N\bigl\{\frac{Y_n^m(\Hat{\mathbf r}_i)}{(2 n+1)a_i} \bigr\}_{0\le|m|\le n}$ of $\mathbf H^1$; respectively, matrix $\mathbb K$ represents operator $\mathring{\mathcal{K}}$.

Figures (2)

  • Figure 1: A system of randomly located spheres of variable radii and dielectric constants.
  • Figure 2: Cubic lattice of non-overlapping spheres.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Remark 3
  • Remark 4
  • ...and 7 more