Stable Recovery of Matrix Gauge Classes from Pointwise Invariants

2026/07/31 by Dexuan Zhou, Huajie Chen, Bernie Hsu +1
Mathematics · Computer Science · #math.NA #cs.NA #msc:15A29 #msc:15A18 #msc:15A21 #msc:05C22 #msc:05C50

paper · pdf

42 pages, 2 figures

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

A parameterized matrix family x↦ H(x) on a configuration domain is determined by its physical content only up to a constant orthogonal change of basis. This gauge ambiguity is intrinsic to data-driven Hamiltonian models, such as tight-binding parameterizations, reduced-order electronic structure methods, or excited-state models. It raises a basic inverse problem: what observations of H(x) suffice to identify the family up to this gauge? The pointwise spectrum is incomplete already for linear families on ℝ. Here, we prove that, under natural non-degeneracy and connectivity assumptions, augmenting the spectrum with loop products of the coupling matrices in the instantaneous eigenframe yields a complete invariant and that inversion is stable. We support the theory with numerical experiments.

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