{
  "abstract": "Beenakker's formula, $U(φ)-U(0)=Δ\\sum_n\\bigl(1-\\sqrt{1-τ_n\\sin^2(φ/2)}\\bigr)$, expresses the zero-temperature energy-phase relation of a short Josephson junction through its normal-state transmission eigenvalues. We generalize this transmission representation beyond the short-junction limit and express the phase-dependent free energy of a time-reversal-symmetric $s$-wave junction as a Matsubara sum over channels with frequency-dependent transmissions. The representation constrains the small-phase expansion of the free energy. In particular, we prove that the normalized ratio of its quartic and quadratic coefficients satisfies $-1\\leα_r\\le7/80$ at finite temperature. We also construct a four-site junction with $α_r>0$, which disproves the conjectured upper bound $α_r\\le0$. So that these results can be trusted without repeating the derivations, every numbered equation is formally verified in the Lean proof assistant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18980",
  "author_total": 1,
  "authors": [
    {
      "name": "Anton R. Akhmerov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frq9dn2mn9m2q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18980",
  "fulltext_url": null,
  "id": "frq9dn2mn9m2q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18980",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "cond-mat.supr-con"
    ],
    [
      "openalex",
      "3104|Physics of Superconductivity and Magnetism"
    ]
  ],
  "openalex": "W7222256383",
  "pdf_url": "https://arxiv.org/pdf/2610.18980",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frq9dn2mn9m2q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:31Z",
      "native_id": "2610.18980",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18980"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Transmission representation of the Josephson free energy beyond the short-junction limit",
  "title_text": "Transmission representation of the Josephson free energy beyond the short-junction limit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:31Z",
  "url": "https://research.draconitic.com/paper/frq9dn2mn9m2q",
  "venue": "arXiv",
  "versions": []
}