Eigenfunctions, free boundaries, and time-frequency localization

[2607.21590] Eigenfunctions, free boundaries, and time-frequency localization Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2607.21590 (math) [Submitted on 23 Jul 2026 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: Eigenfunctions, free boundaries, and time-frequency localization Authors: João P. G. Ramos View PDF HTML (experimental) Abstract: We develop an inverse theory for time--frequency localization operators, formulated as a free-boundary problem in which the localization domain is unknown and its boundary is recovered from prescribed spectral data. The guiding principle is that an eigenfunction can itself be regarded as geometric data determining a localization domain, with strong consequences for the associated variational problem. We obtain five main results. First, for every polynomial $f_0$ sufficiently close to the Gaussian and every $\lambda\in(0,1)$, we construct a real-analytic domain $U_\lambda$ such that $f_0$ is an eigenfunction of the corresponding localization operator with eigenvalue $\lambda$, providing a general inverse construction of localization domains. Second, we recover the null-set invariant Abreu--D"orfler characterization of disks as the only simply connected localization domains having a Hermite polynomial as eigenfunction. Third, we show that simple connectivity is essential: a weighted Hele--Shaw flow yields an infinite-dimensional family of non-radial, doubly connected analytic domains for which the Gaussian is an eigenfunction. Fourth, we prove the optimality of the exponent $1/2$ in the quantitative stability inequality of G'omez--Guerra--Ramos--Tilli. Finally, we prove that every local maximizer of the Gaussian Faber--Krahn problem is a disk and, using a Fock-space concentration--compactness profile decomposition, obtain a new proof of the Nicola--Tilli concentration theorem. Comments: Version 2: 76 pages, one result added, typos corrected Subjects: Analysis of PDEs (math.AP) ; Classical Analysis and ODEs (math.CA); Complex Variables (math.CV); Functional Analysis (math.FA) Cite as: arXiv:2607.21590 [math.AP] (or arXiv:2607.21590v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2607.21590 Focus to learn more arXiv-issued DOI via DataCite Submission history From: João Pedro Ramos [ view email ] [v1] Thu, 23 Jul 2026 17:59:45 UTC (79 KB) [v2] Wed, 26 Aug 2026 20:36:15 UTC (86 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-07 Change to browse by: math math.CA math.CV math.FA 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

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