A spectral theory of quantum charge screening in dense plasmas

A spectral theory of quantum charge screening is developed directly from the self-consistent eigenmodes of the multistream Schrödinger–Poisson (SP) system. Unlike susceptibility-based descriptions of screening, the present formulation derives the dielectric response from the coupled SP eigenproblem and yields an exact two-screening-wavenumber factorization. The spectral evolution of these wavenumbers produces three distinct screening regimes, corresponding to two real poles, a critical double pole, and a complex-conjugate pole pair, for which closed-form real-space screening potentials are obtained. Pole coalescence defines the transition between these regimes and yields a universal critical screening profile at the double-pole point. Finite-temperature screening is formulated as an occupation-weighted spectral average of SP momentum eigenstates, establishing a direct connection between the microscopic spectral states and the macroscopic screened potential. Extension to electron–ion plasmas produces a two-branch collective spectrum with separate contributions to the screening response. The resulting static response is related to quantum-hydrodynamic and quantum-kinetic descriptions while retaining explicit spectral pole information. Finally, the formulation is extended to relativistic screening within the square-root Klein–Gordon–Poisson model.