Zero-energy scattering and the real Bers image on the line

[2602.07373] Zero-energy scattering and the real Bers image on the line Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2602.07373 (math) [Submitted on 7 Feb 2026 ( v1 ), last revised 26 Aug 2026 (this version, v3)] Title: Zero-energy scattering and the real Bers image on the line Authors: Hy P. G. Lam View PDF HTML (experimental) Abstract: Let $\mathrm{Diff}_{\mathcal S}(\mathbb R)$ be the group of orientation-preserving diffeomorphisms $\varphi$ of the line with $\varphi'-1$ Schwartz and $\varphi(x)-x\to0$ as $x\to-\infty$, and let $\beta(\varphi)=\frac12S(\varphi)$ be half the Schwarzian derivative. We determine the image of $\beta$. The coordinate $w_\varphi=\frac12(\log\varphi')'$ identifies $\mathrm{Diff}_{\mathcal S}(\mathbb R)$ with the zero-mean hyperplane $\mathcal S_0(\mathbb R;\mathbb R)$ and turns $\beta$ into $w\mapsto w'-w^2$, so the question is which real Schwartz $q$ equal $w'-w^2$ for some $w$ of zero mean. Nonnegativity of the Schrodinger operator $H_q=-\partial_x^2-q$ decides which $q$ admit such a $w$ at all, and says nothing about the mean of $w$. The mean is a scattering invariant. Let $T_q$ and $R_q$ be the transmission and reflection coefficients of $H_q$. For every real $w\in\mathcal S(\mathbb R)$, with $q=w'-w^2$, we prove $T_q(0)=\mathrm{sech}(\int_{\mathbb R} w\,dx)$ and $R_q(0)=-\tanh(\int_{\mathbb R} w\,dx)$, so $\int_\mathbb R w\,dx$ is read off the scattering matrix at zero energy. Thus $q\in\beta(\mathrm{Diff}_{\mathcal S}(\mathbb R))$ if and only if $H_q$ has no negative eigenvalues and $R_q(0)=0$, equivalently $T_q(0)=1$, and $\varphi\mapsto R_{\beta(\varphi)}$ is a bijection onto an explicit set of Schwartz reflection coefficients. We also compute the differential of $\beta$, whose range depends on the ambient topology. On the Schwartz space, that range is closed and split of codimension two, with normal functionals the first variations of the Wronskian of the two zero-energy Jost solutions and of $R_q(0)$. In every $W^{k,1}$ realization the second functional is unbounded. The range is then a dense proper subspace of the kernel of the first, hence neither closed nor split, and the operator has no bounded inverse on its range. Comments: 14 pages. Corrects and supersedes v1-v2. Title changed Subjects: Differential Geometry (math.DG) ; Analysis of PDEs (math.AP); Functional Analysis (math.FA); Spectral Theory (math.SP) MSC classes: 34L25, 34L40, 53A55, 58D05 Cite as: arXiv:2602.07373 [math.DG] (or arXiv:2602.07373v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2602.07373 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hy Lam [ view email ] [v1] Sat, 7 Feb 2026 05:33:23 UTC (109 KB) [v2] Thu, 26 Feb 2026 02:36:33 UTC (47 KB) [v3] Wed, 26 Aug 2026 20:25:23 UTC (14 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-02 Change to browse by: math math.AP math.FA math.SP References & Citations NASA ADS 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) Major funding support from

Related documents

• Zero-energy scattering and the real Bers image on the line#682574
• A Scattering Result Around a Nonlocalized Equilibria for the Quintic Hartree Equation for Random Fields#314969
• Constraints on Elastic Neutrino Nucleus Scattering in the Fully Coherent Regime from the CONUS Experiment#1053527
• Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction#1106214
• Scattering for the one dimensional Hartree Fock equation#314965