King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density

We study King function arising as radial kernels in the laboratory-frame spherical harmonic expansion of shifted Gaussian distributions. We first clarify their relation with the co-moving Laguerre hierarchy by means of a King--Laguerre expansion. We then derive the King differential equation and show that the associated self-adjoint operator in a Gaussian-weighted Hilbert space is unitarily equivalent to the free radial Schr\"odinger operator on the half-line. This yields the spectral representation and generalized eigenfunction. Finally, we prove that real-parameter King function, lies in the resolvent set, form a dense non-orthogonal system in a natural radial velocity space, providing an approximation-theoretic basis for King mixture representations. Weighted \(L^1\)-integrability criteria and closed-form moment formulas are also derived, justifying the normalization of King function.