一般自伴族有序特征值的 Morse 不等式
Morse inequalities for ordered eigenvalues of generic self-adjoint families
AI总结：
本文针对光滑自伴矩阵族有序特征值的临界点计数问题，提出广义 Morse 族与拓扑临界点概念，证明其通有性并给出 Morse 多项式局部贡献公式，揭示贡献仅依赖于特征值重数和相对位置。
摘要：
在许多应用问题中，人们试图识别并计数光滑参数化自伴矩阵族的某个特定特征值的临界点，参数空间通常是已知且简单的，例如环面。这类问题出现的典型场景包括周期 Schrödinger 算子的 Floquet–Bloch 分解、量子化学中势能面的拓扑、谱优化问题（如流形的最小谱划分）以及图特征函数的节点统计。与处理光滑函数的经典 Morse 理论不同，自伴矩阵族的特征值在对应于重特征值的点处不光滑，这些点根据应用和参数空间维数的不同被称为 diabolical/Dirac/Weyl 点或锥形交叉点。本文建立了一种将 Morse 多项式与特征值重数点关联起来的方法；它利用族的光滑性和自伴性假设给出具体答案。特别地，我们定义了非退化拓扑临界点和广义 Morse 族的概念，证明了广义 Morse 族在适当意义下是通有的，建立了临界性的微分一阶条件，并计算了拓扑临界点对 Morse 多项式的局部贡献。值得注意的是，对 Morse 多项式的非光滑贡献仅取决于特征值重数的大小以及所关注特征值的相对位置，而与算子族的具体细节无关；该贡献用 Grassmann 流形的同调来表示。
英文摘要：
In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet--Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians.
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