Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation
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Quantum Physics arXiv:2601.20873 (quant-ph) [Submitted on 18 Jan 2026] Title:Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation Authors:P.J. Rijken, Th.A. Rijken View a PDF of the paper titled Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation, by P.J. Rijken and Th.A. Rijken View PDF HTML (experimental) Abstract:Eigenvalues of the Breit equation, in which only the static Coulomb potential is considered, have been found. Over the past decades several authors have analyzed the Breit equation to obtain numerically or by approximation an estimation of the energy levels. Various approaches have been used and no determination of the energy levels currently exists that is directly based on the second order Heun differential equation derived. The aim of this work is to provide a method of calculation that can be used to numerically calculate the energy levels for various spin states to high accuracy. From the Breit equation, we derive the corresponding second-order Heun differential equation and continued fraction from which the eigenvalues can be determined very accurately. Next, we present a novel method based on the Green function method, which leads to a semi-infinite determinant from which we are able to obtain the numerical values of the eigenvalues by direct calculation. Using suitable numerical methods for the direct calculation of the continued fraction and the semi-infinite determinant, we show that both methods are consistent within 25 digits of accuracy. We show that the correct energy levels for the Dirac equation follow from our results by a suitable mapping of the variables. The results are in total agreement with earlier calculations found in the literature and extend this by several digits of additional accuracy. The condition on the determinant giving the energy levels provides a rich structure that is promising in extending the results of this work. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2601.20873 [quant-ph] (or arXiv:2601.20873v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.20873 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Thomas Rijken A. [view email] [v1] Sun, 18 Jan 2026 17:32:03 UTC (256 KB) Full-text links: Access Paper: View a PDF of the paper titled Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation, by P.J. Rijken and Th.A. RijkenView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)