Energy levels of the second-harmonic Hamiltonian at large photon numbers

[2608.02788] Energy levels of the second-harmonic Hamiltonian at large photon numbers Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Quantum Physics arXiv:2608.02788 (quant-ph) [Submitted on 3 Aug 2026 ( v1 ), last revised 11 Sep 2026 (this version, v3)] Title: Energy levels of the second-harmonic Hamiltonian at large photon numbers Authors: Boulat Nougmanov View a PDF of the paper titled Energy levels of the second-harmonic Hamiltonian at large photon numbers, by Boulat Nougmanov View PDF HTML (experimental) Abstract: We study the energy spectrum of the degenerate $\chi^{(2)}$ Hamiltonian describing second-harmonic generation and parametric down-conversion at large total excitation numbers. Owing to the conservation of the total excitation number, the spectral problem reduces to the diagonalization of finite-dimensional tridiagonal matrices. Following the approach of Alvarez and Alvarez-Estrada, we map this problem onto an effective one-dimensional Schrödinger equation with a double-well potential. We then derive an implicit quantization condition for the energy levels near the top of the potential barrier and obtain two explicit asymptotic formulas applicable in complementary spectral regions. The ranges of applicability of the resulting formulas are established, and their accuracy is compared with exact matrix diagonalization and with the conventional JWKB approximation. The proposed approach provides an accurate analytical description of the energy levels near the center of the spectrum, where the conventional approximation loses accuracy. Comments: 12 pages, 7 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2608.02788 [quant-ph] (or arXiv:2608.02788v3 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.02788 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Boulat Nougmanov [ view email ] [v1] Mon, 3 Aug 2026 18:41:16 UTC (774 KB) [v2] Fri, 14 Aug 2026 19:31:09 UTC (774 KB) [v3] Fri, 11 Sep 2026 16:32:56 UTC (788 KB) Full-text links: Access Paper: View a PDF of the paper titled Energy levels of the second-harmonic Hamiltonian at large photon numbers, by Boulat Nougmanov View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2026-08 References & Citations INSPIRE HEP NASA ADS Google 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? ) 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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