Hamiltonian-Type Spectral Functionals for Time-Dependent Perturbed Hodge--de Rham Operators
Abstract
Let gt be a smooth family of Riemannian metrics on an oriented manifold of even dimension n≥4, and let Dt=d+δgt+Ψt on the fixed exterior bundle, with Ψt self-adjoint. We evaluate a scalar-weighted residue functional obtained from the operator expression in Hawkins' Hamiltonian action using the coefficient derivative ∂tDt. The velocity contribution depends only on ∂tgt. 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.
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BibTeX
@misc{wei2026hamiltoniantype,
title = {{Hamiltonian-Type Spectral Functionals for Time-Dependent Perturbed Hodge--de Rham Operators}},
author = {Sining Wei and Yong Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25676},
eprint = {2609.25676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25676},
}
RIS
TY - UNPB
TI - Hamiltonian-Type Spectral Functionals for Time-Dependent Perturbed Hodge--de Rham Operators
AU - Sining Wei
AU - Yong Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25676
UR - https://arxiv.org/abs/2609.25676
AB - Let g_t be a smooth family of Riemannian metrics on an oriented manifold of even dimension n≥4, 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 ∂_tD_t. The velocity contribution depends only on ∂_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.
ER -