A Sharp Spectral Mantel Theorem for Quantum Transport
Abstract
We study the maximum average quantum transition probability between distinct vertices of a triangle-free graph, equivalently the minimum average return probability, at times inversely proportional to the order. For every compact interval of positive scaled times below τc≈3.79099, the balanced complete bipartite graph is the unique maximizer for all sufficiently large orders. The threshold is sharp: it solves τc=4sin(τc/2), and the balanced complete bipartite graph is eventually suboptimal at every larger scaled time. The corresponding graphon functional combines the edge density with alternating even cycle densities. We prove that the balanced complete bipartite graphon is its unique maximizer through the critical time. On a nonempty interval immediately afterward, the unique maximizer is a balanced bipartite graphon with constant cross-edge weight below one. After normalization by the squared time, we obtain uniform square-root L1 stability through the critical endpoint and at zero time. We also solve the bipartite problem at every positive time and establish sharp cut distance stability. The proofs combine spectral interpolation, an exactly certified six-vertex moment inequality, vertex-measure variation, and a minimum-degree argument. A uniform finite approximation bound and vertex cloning connect the graphon and exact finite results.
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BibTeX
@misc{song2026sharp,
title = {{A Sharp Spectral Mantel Theorem for Quantum Transport}},
author = {Xingkun Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27235},
eprint = {2608.27235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27235},
}
RIS
TY - UNPB
TI - A Sharp Spectral Mantel Theorem for Quantum Transport
AU - Xingkun Song
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.27235
UR - https://arxiv.org/abs/2608.27235
AB - We study the maximum average quantum transition probability between distinct vertices of a triangle-free graph, equivalently the minimum average return probability, at times inversely proportional to the order. For every compact interval of positive scaled times below τ_c≈3.79099, the balanced complete bipartite graph is the unique maximizer for all sufficiently large orders. The threshold is sharp: it solves τ_c=4sin(τ_c/2), and the balanced complete bipartite graph is eventually suboptimal at every larger scaled time. The corresponding graphon functional combines the edge density with alternating even cycle densities. We prove that the balanced complete bipartite graphon is its unique maximizer through the critical time. On a nonempty interval immediately afterward, the unique maximizer is a balanced bipartite graphon with constant cross-edge weight below one. After normalization by the squared time, we obtain uniform square-root L^1 stability through the critical endpoint and at zero time. We also solve the bipartite problem at every positive time and establish sharp cut distance stability. The proofs combine spectral interpolation, an exactly certified six-vertex moment inequality, vertex-measure variation, and a minimum-degree argument. A uniform finite approximation bound and vertex cloning connect the graphon and exact finite results.
ER -