2026/07/16 by Baruch Schneider, Diana Barseghyan Schneiderová, Yifan Zhang
#math.SP
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 H JH=J. For m>0 we consider \mathcal Am,H = H-1(-iJ\frac ddt+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 ∫\rm 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-1/2 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.