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

[2601.10954] Exact Spectrum of a Curvature-Adapted $\mathbb{Z}_{2}^{3}$ Dunkl--Deng--Fan/Eckart System Skip to main content Search Submit Donate Log in Mathematical Physics arXiv:2601.10954 (math-ph) [Submitted on 16 Jan 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Exact Spectrum of a Curvature-Adapted $\mathbb{Z}_{2}^{3}$ Dunkl--Deng--Fan/Eckart System Authors: Nikko John Leo S. Lobos View PDF HTML (experimental) Abstract: 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 $\alpha=2\kappa$ 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 $\eta*{n\ell}^{2}<q_{\kappa}/2$. 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. Comments: 22 pages Subjects: Mathematical Physics (math-ph) Cite as: arXiv:2601.10954 [math-ph] (or arXiv:2601.10954v2 [math-ph] for this version) https://doi.org/10.48550/arXiv.2601.10954 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1088/1402-4896/ae9b49 https://doi.org/10.1088/1402-4896/ae9b49 Focus to learn more DOI(s) linking to related resources Submission history From: Nikko John Leo Lobos [ view email ] [v1] Fri, 16 Jan 2026 02:47:07 UTC (49 KB) [v2] Thu, 27 Aug 2026 01:36:56 UTC (27 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2026-01 Change to browse by: math math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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