Exact Spectrum of a Curvature-Adapted ${\mathbb{Z}}_2^3$ Dunkl--Deng--Fan/Eckart System

Abstract

This study constructs and exactly solves a curvature-adapted radial Dunkl Hamiltonian for the coordinate-reflection group $\mathbb{Z}\ast 2^3$ 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 $\alpha =2\kappa$ 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 $\eta \ast {n\ell }^2<q_{\kappa }/2$. Dunkl multiplicities affect the radial spectrum only through $\gamma ={\mu }_1+{\mu }_2+{\mu }_3$, whereas reflection eigenvalues restrict the angular degrees to $\ell =p+2k$, with total multiplicity $2\ell +1$. On each surviving branch, energy increases with $\ell$ and $\gamma$, while the number of bound states decreases. Quadratic-form Hellmann--Feynman identities give exact hyperbolic-interaction expectation values, and adjacent-level spacings follow directly from the spectrum. The undeformed and correlated flat limits recover the ordinary curvature-matched Eckart and Dunkl--Kratzer spectra, respectively. Finite-difference calculations for $\gamma =0$ and $1/2$ confirm the energies, continuum thresholds, and state counts. This tuned curvature-adapted construction is not the generic Euclidean Dunkl--Deng--Fan problem. It isolates the effects of negative curvature and reflection deformation on level ordering, binding thresholds, and bound-state counts, and provides analytic benchmarks for numerical and approximate treatments of singular curvature- and reflection-deformed radial Hamiltonians.

22 pages