A Variational Approach to Spectral Shift and Spectral Minimal Partitions

谱位移和谱最小划分的变分方法

• 批准号: 2247473
• 负责人: GREGORY BERKOLAIKO
• 金额: $ 27.62万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247473&HistoricalAwards=false
• 关键词: Variational; Approach; Spectral; Shift; Minimal

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.

自伴算子参数族的特征值优化是应用科学中各种问题的数学抽象。例如，量子化学中的势能面是本征值与原子在分子中位置的依赖关系的图景；在这个图景中，电子寻找最低点，迫使分子进入相应的构型。对反应扩散系统中图案形成的研究类似地可以导致优化可用空间分区的能量(特征值)的问题；这个问题被称为“谱最小分区问题”。这两个示例的共同特点是参数数量非常多。本项目的基本思想是，当参数数目足够大时，可以从关于本征值行为的局部信息中获得关于算子族的全局信息。这样做的一个实际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。这个项目的问题将被用来指导研究生和本科生的研究人员，并将创建和维护致力于度量图上的谱分析的WikiBook。该项目中要解决的具体数学问题是对边界条件的特征值依赖，它直接与‘谱最小划分问题’中划分边界的位置有关。在算子论的水平上，这将被表示为固定对称算子的自伴扩张的变体。其目的是证明临界点上本征值的莫尔斯指数与算子相对于参考算子的谱移位之间的联系。由于Morse指数描述了本征值相对于边界条件中的扰动的局部稳定性，因此这种联系将允许以谱移位的形式获得全局信息。频谱漂移的知识将导致能量格局的界限，而这又将证明局部最小值是全局最优的。用于建立联系的工具将包括拉格朗日·格拉斯曼、Krein-Naimark预解公式、Dirichlet-to-Neumann映射和Maslov指数技术的自伴扩展的参数化。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。