Exact Screening-Ranged Expansions for Many-Body Electrostatics

We present an exact many-body framework for electrostatic interactions among $N$ arbitrarily charged spheres in an electrolyte, modeled by the linearized Poisson–Boltzmann equation. Building on a spectral analysis of nonstandard Neumann–Poincaré-type operators introduced in a companion mathematical work arXiv:2512.08684, we construct convergent screening-ranged series for the potential, interaction energy, and forces, where each term is associated with a well-defined Debye–Hückel screening order and can be obtained evaluating an analytical expression rather than numerically solving an infinitely dimensional linear system. This formulation unifies and extends classical and recent approaches, providing a rigorous basis for electrostatic interactions among heterogeneously charged particles (including Janus colloids) and yielding many-body generalizations of analytical explicit-form results previously available only for two-body systems. The framework captures and clarifies complex effects such as asymmetric dielectric screening, opposite-charge repulsion, and like-charge attraction, which remain largely analytically elusive in existing treatments. Beyond its fundamental significance, the method leads to numerically efficient schemes, offering a versatile tool for modeling colloids and soft/biological matter in electrolytic solution.