The continuous spectrum of bound states in expulsive potentials
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Quantum Physics arXiv:2602.07281 (quant-ph) [Submitted on 7 Feb 2026] Title:The continuous spectrum of bound states in expulsive potentials Authors:H. Sakaguchi, B.A. Malomed, A.C. Aristotelous, E.G. Charalampidis View a PDF of the paper titled The continuous spectrum of bound states in expulsive potentials, by H. Sakaguchi and 3 other authors View PDF HTML (experimental) Abstract:On the contrary to the common intuition that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schrödinger equations, which include expulsive potentials that are \emph{steeper than the quadratic} (anti-harmonic-oscillator) ones, give rise to \emph{normalizable} (effectively localized) eigenstates. These states constitute full continuous spectra in the 1D and 2D cases alike. In 1D, these are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic approximations for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with numerically found counterparts. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability. On the other hand, the quintic self-focusing term, which occurs in the photonic version of the 1D model, initiates the dynamical collapse of states whose norm exceeds a critical value. Comments: Subjects: Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics) Cite as: arXiv:2602.07281 [quant-ph] (or arXiv:2602.07281v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.07281 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Efstathios Charalampidis [view email] [v1] Sat, 7 Feb 2026 00:10:49 UTC (407 KB) Full-text links: Access Paper: View a PDF of the paper titled The continuous spectrum of bound states in expulsive potentials, by H. Sakaguchi and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: cond-mat cond-mat.quant-gas nlin nlin.PS physics physics.optics 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?)