A Variational Approach to Spectral Shift and Spectral Minimal Partitions

自伴随算子参数族的特征值优化是对应用科学中出现的各种问题的数学抽象。例如，量子化学中的势能面是特征值对分子内原子位置的依赖性的景观；在这种情况下，电子会寻找最低点，迫使分子形成相应的构型。研究反应-扩散系统的模式形成同样可以导致可用空间分区的能量（特征值）的优化问题；这个问题被称为“谱最小分割问题”。这两个例子的共同特点是参数非常多。本项目的基本思想是，当参数数量足够大时，可以从有关特征值行为的局部信息中获得有关算子族的全局信息。这样做的一个实际的重要结果是能够证明实验发现的局部最小值是全局最优的。这个项目的问题将用于指导研究生和本科生的研究人员，并将创建和维护一个专门用于度量图光谱分析的维基教科书。该项目中要解决的具体数学问题是特征值对边界条件的依赖，这与“谱最小划分问题”中分区边界的位置直接相关。在算子理论层面上，这将被表示为一个固定对称算子的自伴随扩展的变分。目的是证明在临界点处特征值的莫尔斯指数和算子相对于参考算子的谱移之间的联系。由于莫尔斯指数描述了特征值相对于边界条件下的扰动的局部稳定性，这个链接将允许人们以谱移的形式获得全局信息。谱移的知识将导致能源格局的边界，反过来，将证明局部最小值是全局最优的。用于建立这种联系的工具将包括拉格朗日格拉斯曼扩展的参数化、克莱恩-奈马克解析公式、狄利克雷-诺伊曼映射和马斯洛夫指数技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Eigenvalue optimization for a parametric family of self-adjoint operators is a mathematical abstraction of a wide variety of questions arising in applied science. For example, the potential energy surface in quantum chemistry is the landscape of the dependence of an eigenvalue on the atoms' positions within the molecule; in this landscape, electrons seek out the lowest point, forcing the molecule into the corresponding configuration. Study of pattern-formation in reaction-diffusion systems can similarly lead to the question of optimizing energy (eigenvalue) of a partition of the available space; this question is known as the ‘spectral minimal partition problem’. The common feature of these two examples is a very large number of parameters. The underlying idea of the present project is that when the number of parameters is sufficiently large, it is possible to obtain global information about the operator family from the local information about the eigenvalue behavior. A practically important consequence of this is the ability to certify an experimentally found local minimum as being globally optimal. Questions from this project will be used to mentor graduate and undergraduate researchers, and a WikiBook devoted to spectral analysis on metric graphs will be created and maintained.The specific mathematical question to be addressed in the project is the eigenvalue dependence on the boundary conditions, which is directly related to the location of the partition boundaries in the ‘spectral minimal partition problem’. On the operator-theoretic level, this will be expressed as variation of the self-adjoint extension of a fixed symmetric operator. The goal is to prove a link between the Morse index of the eigenvalue at a critical point and the spectral shift of the operator with respect to a reference operator. Since the Morse index describes local stability of the eigenvalue with respect to perturbations in the boundary conditions, this link will allow one to obtain global information in the form of the spectral shift. The knowledge of spectral shift will then lead to bounds on the energy landscape which, in turn, will certify the local minimum as being globally optimal. The tools used for establishing the link will include parametrization of self-adjoint extensions by the Lagrangian Grassmannian, the Krein-Naimark resolvent formula, Dirichlet-to-Neumann map and the Maslov index techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.