Poisson operator on the interacting Fock space associated with a discrete-time quantum walk

[2607.29345] Poisson operator on the interacting Fock space associated with a discrete-time quantum walk Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2607.29345 (math) [Submitted on 31 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Poisson operator on the interacting Fock space associated with a discrete-time quantum walk Authors: Daiju Funakawa , Yuki Ueda , Kazuyuki Wada View PDF HTML (experimental) Abstract: We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory. More precisely, we compute the moment-generating function and moments of the QW-Poisson operator, and obtain a limit theorem for the Konno distribution via a Poisson approximation. We also investigate the Boolean self-decomposability of the Konno distribution and the shifted reversed QW-Poisson distribution. Comments: 40 pages. v3: We revised the atom at zero of the QW-Poisson distribution (Theorem 1.3) and reorganized the structure of the paper Subjects: Functional Analysis (math.FA) ; Mathematical Physics (math-ph); Operator Algebras (math.OA); Probability (math.PR) MSC classes: 46L53, 46L54, 46L60, 47B36, 47B93, 60F05 Cite as: arXiv:2607.29345 [math.FA] (or arXiv:2607.29345v3 [math.FA] for this version) https://doi.org/10.48550/arXiv.2607.29345 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yuki Ueda [ view email ] [v1] Fri, 31 Jul 2026 12:29:47 UTC (28 KB) [v2] Mon, 3 Aug 2026 12:15:59 UTC (28 KB) [v3] Thu, 27 Aug 2026 06:16:34 UTC (28 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-07 Change to browse by: math math-ph math.MP math.OA math.PR 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. 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? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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