Hamiltonian型谱泛函用于含时扰动的Hodge--de Rham算子
Hamiltonian-Type Spectral Functionals for Time-Dependent Perturbed Hodge--de Rham Operators
- School of Data Science and Artificial Intelligence Dongbei University of Finance and Economics（东北财经大学数据科学与人工智能学院）
- School of Mathematics and Statistics Northeast Normal University（东北师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对含时扰动的Hodge--de Rham算子，构造Hamiltonian型谱泛函，通过符号计算化简边界项，在翘曲领口紧流形上显式求边界留数，并给出空边界时的闭式公式。
AI中文摘要：
设$g_t$是偶数维$n\geq4$的定向流形上黎曼度量的光滑族，并设$D_t=d+\delta_{g_t}+\Psi_t$作用于固定的外微分丛上，其中$\Psi_t$为自伴算子。我们利用系数导数$\partial_tD_t$，对Hawkins哈密顿作用量中的算子表达式求出一个标量加权留数泛函。速度贡献仅依赖于$\partial_tg_t$。直接的符号计算将时滞修正化简为散度项和Dirichlet项；对于非恒定测试函数，分部积分保留一个混合梯度项。对于具有翘曲领口的紧流形，我们指定了加权KKW项与交换子修正的相容分解。它们的边界留数被显式计算。在加权KKW分解中，领口导数与扰动贡献相互抵消，剩余的边界项涉及标量权重的法向导数。当边界为空时，恢复闭式公式。
英文摘要：
Let $g_t$ be a smooth family of Riemannian metrics on an oriented manifold of even dimension $n\geq4$, and let $D_t=d+δ_{g_t}+Ψ_t$ on the fixed exterior bundle, with $Ψ_t$ self-adjoint. We evaluate a scalar-weighted residue functional obtained from the operator expression in Hawkins' Hamiltonian action using the coefficient derivative $\partial_tD_t$. The velocity contribution depends only on $\partial_tg_t$. A direct symbol calculation reduces the lapse correction to a divergence and a Dirichlet term; for a nonconstant test function, integration by parts retains a mixed gradient term. For compact manifolds with a warped collar, we specify compatible factorizations of the weighted KKW term and of the commutator correction. Their boundary residues are computed explicitly. The collar-derivative and perturbation contributions cancel in the weighted KKW factorization, and the remaining boundary terms involve normal derivatives of the scalar weights. The closed formula is recovered when the boundary is empty.
