发表机构
Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限图 $p$-拉普拉斯算子的完全谱有限，解决 Amghibech 猜想，并给出谱基数与拓扑量的指数界，揭示 $p=2$ 处的转变及指数增长率为 3 的极值现象。
AI 中文摘要
我们证明对于每个有限图和每个实数 $p>1$，图 $p$-拉普拉斯算子的完全谱是有限的。这解决了 Amghibech (2003) 提出的完全图谱有限性问题。更一般地，对于具有任意实势的带符号加权图 $p$-薛定谔算子，我们获得了以顶点数和正权边数为界的不等式，这些界对所有系数和 $p$ 一致成立。在具有 $n$ 个顶点和 $m$ 条正权边的图上，谱基数和对数以及归一化特征向量集的连通分量总数的对数均为 $O((n+m)^2)$；其有理贝蒂数在完全谱上的总和至多为 $\exp(C(n+m)^2)$，其中 $C$ 为绝对常数。这些拓扑界推广到由任意有限族线性形式构建的齐次特征问题，包括矩阵对的广义 $p$-特征值问题。对于均匀加权的 $K_n$ 且 $p\ne2$，我们将非恒定特征线识别为 $\mathbb RP^{n-2}$ 的坐标超平面胞腔分解的重心，并确定它们在 $p=2$ 两侧的局部莫尔斯数据。重整化对数极限描述了 $p=2$ 处的转变。正整数权重随后分离这些临界值，对于每个固定的 $p\ne2$ 和每个 $n\geq2$，给出至少 $(3^n-2^{n+1}+3)/2$ 个不同的谱值。相比之下，$p=2$ 时的最大谱基数为 $n$；在 $p=4$ 时，无符号类和一般类的最大谱基数均具有精确为 $3$ 的指数增长率。对于 $n\geq3$，相同的例子解决了 Amghibech 的极值问题。证明结合了 o-极小平坦几何与射影 $L^p$-对偶、莫尔斯理论、临界群和张量特征值界。
英文摘要
We prove that the full spectrum of the graph $p$-Laplacian is finite for every finite graph and every real $p>1$. This resolves the problem of finiteness of the full graph spectrum posed by Amghibech (2003). More generally, for signed weighted graph $p$-Schrödinger operators with arbitrary real potentials, we obtain bounds in terms of the numbers of vertices and positive-weight edges, uniform in all coefficients and in $p$. On $n$ vertices with $m$ positive-weight edges, the logarithms of the spectral cardinality and of the total number of connected components of the normalised eigenvector sets are $O((n+m)^2)$; the sum over the full spectrum of their rational Betti numbers is at most $\exp(C(n+m)^2)$ for an absolute constant $C$. These topological bounds extend to homogeneous eigenproblems built from arbitrary finite families of linear forms, including generalised $p$-eigenvalue problems for matrix pairs. For the uniformly weighted $K_n$ and $p\ne2$, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of $\mathbb RP^{n-2}$ and determine their local Morse data on either side of $p=2$. A renormalised logarithmic limit describes the transition at $p=2$. Positive integer weights then separate these critical values, giving at least $(3^n-2^{n+1}+3)/2$ distinct spectral values for every fixed $p\ne2$ and every $n\geq2$. By contrast, the maximal spectral cardinality at $p=2$ is $n$; at $p=4$, the maximal spectral cardinalities in the unsigned and general classes both have exponential growth rate exactly $3$. For $n\geq3$ the same examples resolve Amghibech's extremal question. The proof combines o-minimal and Pfaffian geometry with projective $L^p$-duality, Morse theory, critical groups, and tensor eigenvalue bounds.
