Laplace--King representations: density and spectral theory
Abstract
Laplace--King representations combine spherical harmonics with King functions [Wang et al., Chin. Phys. B \textbf{34}, 065201 (2025)], the radial kernels of shifted isotropic Gaussians. The radial parameters may vary across angular modes. We prove that, for every angular degree, fixed-width kernels with positive real shifts have dense complex linear span in a Gaussian-weighted radial L2 space. Finite Laplace--King representations are dense in the corresponding three-dimensional weighted space; allowing variable widths preserves density in the same reference norm. A generating identity connects the kernels to generalized Laguerre polynomials. The self-adjoint King operator is unitarily equivalent to the free radial Schrödinger operator; its spectral resolution defines a continuous King mixture model (KMM) through imaginary-shift kernels in a distinct weighted Hilbert space.
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BibTeX
@misc{wang2026laplaceking,
title = {{Laplace--King representations: density and spectral theory}},
author = {Yanpeng Wang and Zhe Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.12455},
eprint = {2606.12455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12455},
}
RIS
TY - UNPB
TI - Laplace--King representations: density and spectral theory
AU - Yanpeng Wang
AU - Zhe Gao
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2606.12455
UR - https://arxiv.org/abs/2606.12455
AB - Laplace--King representations combine spherical harmonics with King functions [Wang et al., Chin. Phys. B \textbf{34}, 065201 (2025)], the radial kernels of shifted isotropic Gaussians. The radial parameters may vary across angular modes. We prove that, for every angular degree, fixed-width kernels with positive real shifts have dense complex linear span in a Gaussian-weighted radial L^2 space. Finite Laplace--King representations are dense in the corresponding three-dimensional weighted space; allowing variable widths preserves density in the same reference norm. A generating identity connects the kernels to generalized Laguerre polynomials. The self-adjoint King operator is unitarily equivalent to the free radial Schrödinger operator; its spectral resolution defines a continuous King mixture model (KMM) through imaginary-shift kernels in a distinct weighted Hilbert space.
ER -