量子输运中无三角图的一个极值谱问题
A Sharp Spectral Mantel Theorem for Quantum Transport
- School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）
- Qinghai Institute of Applied Mathematics（青海省应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对量子输运相关的无三角图极值谱问题，确定了特定参数区间内图函子的最大值及唯一最大化子，给出了稳定性结果，相关有限极值值收敛到图函子最大值。
AI中文摘要：
对于阶数为n、邻接矩阵为A的图G，设F_G(t)为不同有序顶点对的|(exp(-𝚤tA))_{vu}|²的平均值。在稠密标度t=τ/n下，量n²F_G(τ/n)导出图函子Φ_τ，其主项为τ²乘以边密度，剩余项构成偶圈密度的加权交错级数。对于0≤τ≤τ_c，我们确定了Φ_τ在所有无三角图函子上的精确最大值：当0<τ≤τ_c时，平衡完全二部图函子B₁是唯一最大化子（弱同构意义下），其中τ_c是方程τ_c=4sin(τ_c/2)的唯一正解，τ_c≈3.79099，最大值等于4(1-cos(τ/2))；该阈值是尖锐的，当τ>τ_c时，B₁不再是全局最优。对于τ>τ_c，二部类内的唯一最大化子（弱同构意义下）是平衡二部图函子B_{q_τ}，其中q_τ∈(0,1)，τ>τ_c时的无约束最大化问题仍未解决。我们还证明了显式边密度亏缺界与定量割距离稳定性，在(0,τ_c)的紧子区间上对τ一致，以及在(0,τ_c]的紧子区间上的定性割距离稳定性；对应的有限无三角极值值局部一致收敛到图函子最大值，在[0,τ_c]上误差为O(n⁻¹)。证明使用了谱图函子泛函的系数准则，结合六阶谱下界、四顶点不等式与六顶点矩不等式，后者由精确有理标志代数证书建立。
英文摘要：
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 $τ_{\mathrm c}\approx3.79099$, the balanced complete bipartite graph is the unique maximizer for all sufficiently large orders. The threshold is sharp: it solves $τ_{\mathrm c}=4\sin(τ_{\mathrm c}/2)$, 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.
