从逐点不变量稳定恢复矩阵规范类
Stable Recovery of Matrix Gauge Classes from Pointwise Invariants
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中文总结 AI 辅助
本文针对数据驱动哈密顿模型的固有规范模糊性，提出结合谱与耦合矩阵环乘积的逐点不变量方法，证明其可稳定恢复矩阵规范类，并通过数值实验验证理论。
中文摘要 AI 辅助
构型域上的参数化矩阵族 $x\mapsto H(x)$ 仅通过其物理内容确定，至多相差一个常数正交基变换，这种规范模糊性是数据驱动哈密顿模型（如紧束缚参数化、降阶电子结构方法或激发态模型）的固有特性。它提出了一个基本逆问题：对 $H(x)$ 的何种观测足以在该规范下识别该族？逐点谱对 $\mathbb{R}$ 上的线性族而言已不完整。本文证明，在自然非退化和连通性假设下，将谱与瞬时本征基中耦合矩阵的环乘积相结合可得到完整不变量，且反演是稳定的。我们通过数值实验验证了该理论。
英文摘要
A parameterized matrix family $x\mapsto H(x)$ 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 $\mathbb{R}$. 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.
