Exact Spectrum of a Curvature-Adapted $\mathbb{Z}_{2}^{3}$ Dunkl--Deng--Fan/Eckart System

This study constructs and exactly solves a curvature-adapted radial Dunkl Hamiltonian for the coordinate-reflection group $\mathbb{Z}*{2}^{3}$ on 3D hyperbolic space, determining how curvature and Dunkl multiplicities govern the discrete spectrum and admissible radial states. The warped-product kinetic operator is defined by its Friedrichs extension, which selects the regular boundary condition in singular radial channels. The restriction $α=2κ$ is an exact-solvability condition, not a general relation between molecular range and spatial curvature, and reduces the centrifugal and Deng--Fan terms exactly to a generalized Eckart problem. We derive the finite spectrum, Jacobi-polynomial radial eigenfunctions, normalization integrals, and normalizability condition $η*{n\ell}^{2}