{
  "abstract": "We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik--Adamyan and appearing in the appendix of Alpay--Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\\mathbb C^{2n}$ satisfying $JB=-BJ$, and let $H>0$ satisfy $HJH=J$. For $m>0$ we consider $$ \\mathcal A_{m,H}=H^{-1}\\left(-iJ\\frac{d}{dt}+mB\\right) $$ in the weighted space $L^2_H$. A locally absolutely continuous $J$-unitary gauge $Θ$ representing $H$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient $$ P_{m,Θ}=-iΘ^*JΘ'+m(Θ^*BΘ-B). $$ Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\\int\\mathrm{Tr}|P_{m,Θ}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman--Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $H^{-1/2}H'H^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15504",
  "author_total": 3,
  "authors": [
    {
      "name": "Baruch Schneider"
    },
    {
      "name": "Diana Barseghyan Schneiderová"
    },
    {
      "name": "Yifan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1azs36974zmgq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15504",
  "fulltext_url": null,
  "id": "1azs36974zmgq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15504",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7169816739",
  "pdf_url": "https://arxiv.org/pdf/2607.15504",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1azs36974zmgq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2607.15504",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15504"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians",
  "title_text": "Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1azs36974zmgq",
  "venue": "arXiv",
  "versions": []
}