Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians
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 ℂ2n satisfying JB=−BJ, and let H>0 satisfy HJH=J. For m>0 we consider 𝒜m,H=H−1(−iJd/dt+mB) in the weighted space L2H. A locally absolutely continuous J-unitary gauge Θ representing H reduces this expression to the free massive Dirac operator plus the Hermitian coefficient Pm,Θ=−iΘ*JΘ'+m(Θ*BΘ−B). Whenever this coefficient belongs to L2, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing ∫Tr|Pm,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 L2 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 2nm3/2. At zero mass, the optimal-gauge energy is computed explicitly in terms of H−1/2H'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.
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BibTeX
@misc{schneider2026liebthirring,
title = {{Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians}},
author = {Baruch Schneider and Diana Barseghyan Schneiderová and Yifan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.15504},
eprint = {2607.15504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15504},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians
AU - Baruch Schneider
AU - Diana Barseghyan Schneiderová
AU - Yifan Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2607.15504
UR - https://arxiv.org/abs/2607.15504
AB - 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 ℂ^2n satisfying JB=−BJ, and let H>0 satisfy HJH=J. For m>0 we consider 𝒜_m,H=H^−1(−iJd/dt+mB) 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 ∫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^−1/2H'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.
ER -