Laplace--King representations: density and spectral theory

Laplace–King representations combine spherical harmonics with King functions [Wang et al., Chin. Phys. B $\mathbf{\text{34}}$, 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 Schr"odinger operator; its spectral resolution defines a continuous King mixture model (KMM) through imaginary-shift kernels in a distinct weighted Hilbert space.