{
  "abstract": "We consider the lowest eigenvalue $λ(b)$ of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength $b>0$. We prove that $λ$ is strictly increasing on $(0,+\\infty)$. This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound $λ(b)<Θ_0 b$, where $Θ_0$ is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field $H_{C_3}^{\\mathrm{loc}}$ of Ginzburg--Landau theory is, in the disk, uniquely determined for every value of the Ginzburg--Landau parameter, and not only for large ones. We also give a second proof of the bound $λ(b)<Θ_0 b$, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for $λ(b)$, valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08774",
  "author_total": 2,
  "authors": [
    {
      "name": "Corentin Léna"
    },
    {
      "name": "Mikael Sundqvist"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/81349x3v4hrap.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08774",
  "fulltext_url": null,
  "id": "81349x3v4hrap",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08774",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7212010449",
  "pdf_url": "https://arxiv.org/pdf/2609.08774",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/81349x3v4hrap.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk",
  "title_text": "Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/81349x3v4hrap",
  "venue": "arXiv",
  "versions": []
}