Exact spectrum of a curvature-adapted Z23 Dunkl–Deng–Fan/Eckart system

Abstract

The objective of this study is to construct and solve exactly a curvature-adapted radial Dunkl Hamiltonian associated with the coordinate-reflection group Z23 on three-dimensional hyperbolic space, and to determine how curvature and Dunkl multiplicities control its discrete spectrum and admissible radial states. The warped-product kinetic operator is realized through 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; it reduces the centrifugal and Deng–Fan terms without approximation to a generalized Eckart problem. We obtain the finite spectrum, Jacobi-polynomial radial eigenfunctions, normalization integrals, and strict normalizability condition ηnℓ2<qκ/2. Dunkl multiplicities enter the radial spectrum only through γ=μ1+μ2+μ3; reflection eigenvalues instead restrict the angular degrees to ℓ=p+2k, with total multiplicity 2ℓ+1. Along each surviving branch, the energy increases with ℓ and γ, while the number of bound states decreases. Quadratic-form Hellmann–Feynman identities yield exact hyperbolic-interaction expectation values, whereas adjacent-level spacings follow from the spectrum. The undeformed and correlated flat limits recover the ordinary curvature-matched Eckart and Dunkl–Kratzer spectra, respectively. Finite-difference calculations for γ=0 and 1/2 confirm the energies, continuum thresholds, and state counts. The construction is a tuned curvature-adapted system, not the generic Euclidean Dunkl–Deng–Fan problem. Its physical significance is to isolate, within a controlled solvable model, the effects of negative curvature and reflection deformation on level ordering, binding thresholds, and the number of bound states. The resulting formulas also provide analytic benchmarks for numerical and approximate treatments of singular curvature- and reflection-deformed radial Hamiltonians.