Mathematical Study of King Functions in Shifted Gaussian Distributions and Spectral Theory
Researchers have derived a comprehensive mathematical framework for King functions, which arise as radial kernels in spherical harmonic expansions of shifted Gaussian distributions. The work connects King functions to Laguerre polynomials, establishes a governing differential equation, and proves their density in a natural radial velocity space. These results provide a rigorous theoretical foundation for King mixture representations used in plasma physics modeling.
A new preprint posted to arXiv by Yanpeng Wang develops a detailed mathematical treatment of King functions, special functions that appear when expanding shifted Gaussian distributions in laboratory-frame spherical harmonics. The paper first establishes a King–Laguerre expansion, clarifying how King functions relate to the classical Laguerre polynomial hierarchy in the co-moving frame. A King differential equation is then derived, and the associated self-adjoint operator in a Gaussian-weighted Hilbert space is shown to be unitarily equivalent to the free radial Schrödinger operator on the half-line, yielding an explicit spectral representation and generalized eigenfunctions. The authors further prove that real-parameter King functions, lying in the resolvent set, form a dense but non-orthogonal system in a natural radial velocity space, which underpins their use in approximation theory and mixture representations. Weighted L¹-integrability criteria and closed-form moment formulas are also derived, rigorously justifying the normalization of King functions. The results have direct relevance to plasma physics, where King distributions are used to model particle velocity distributions in non-equilibrium settings.
What's missing
As a preprint, this work has not yet undergone formal peer review. The paper does not discuss numerical validation or convergence rates for practical King mixture approximations, nor does it address how the theoretical framework performs for specific plasma physics applications. Open questions include whether the non-orthogonality of the King system poses numerical challenges in applied settings.
What different sources said
- arXiv physicsCenter
King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density
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