Uniform High-Frequency Localization on Quantum Graphs
Abstract
We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least L2-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant C∞(ω;ℓ) and its singular-completion counterpart Csing(ω;ℓ) satisfy C∞(ω;ℓ)<1/2<Csing(ω;ℓ). Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.
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BibTeX
@misc{nguyen2026uniform,
title = {{Uniform High-Frequency Localization on Quantum Graphs}},
author = {Binh T. Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17623},
eprint = {2609.17623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17623},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Uniform High-Frequency Localization on Quantum Graphs
AU - Binh T. Nguyen
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17623
UR - https://arxiv.org/abs/2609.17623
AB - We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least L^2-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant C_∞(ω;ℓ) and its singular-completion counterpart C_sing(ω;ℓ) satisfy C_∞(ω;ℓ)<1/2<C_sing(ω;ℓ). Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.
ER -