Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series
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Quantum Physics arXiv:2605.11031 (quant-ph) [Submitted on 10 May 2026] Title:Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series Authors:Ramon Moya View a PDF of the paper titled Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series, by Ramon Moya View PDF Abstract:We identify a class of finite quantum systems, namely, acyclic systems whose transition graph is a directed acyclic graph (DAG), for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G_0(E)V. If T^{m+1} = 0, then the exact solution of the Lippmann-Schwinger equation is the finite sum |psi> = sum_{k=0}^{m} T^k |phi>, with no condition on ||T||. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A_4 = t_{42}t_{21} + t_{43}t_{31}, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born-SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric, the Born-SON depth, quantifying the intrinsic complexity of an acyclic quantum system. Comments: Subjects: Quantum Physics (quant-ph) MSC classes: 81U05, 47A10, 05C20, 81Q10, 15A16 Cite as: arXiv:2605.11031 [quant-ph] (or arXiv:2605.11031v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.11031 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ramon Moya Mr. [view email] [v1] Sun, 10 May 2026 22:43:37 UTC (1,029 KB) Full-text links: Access Paper: View a PDF of the paper titled Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series, by Ramon MoyaView PDF view license Current browse context: quant-ph new | recent | 2026-05 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?) 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?)