标题： 奇异拉格朗日系统与贝塞尔型微分算子的指标理论 显示英文标题

标题： Index theory for singular Lagrangian systems and Bessel-type differential operators

摘要： 本文的目的是为奇异拉格朗日系统发展一种指标理论，特别关注由贝塞尔型微分算子给出的重要奇异算子类别。主要动机是解决在广泛的应用中出现的奇异算子带来的若干挑战：天体力学（例如行星运动中的扰动）、具有时间依赖激励的振荡系统、电磁学（如非均匀介质中的波动方程）以及量子力学（特别是某些具有周期势的薛定谔方程）。我们追求两个主要目标。首先，我们建立奇异斯特姆-刘维尔算子的间隙连续路径的谱流公式和莫尔斯指标定理。通过这些指标公式，我们为一大类贝塞尔型微分算子构建了莫尔斯指标理论，并将其应用于引力n体问题的一族渐近解。最后，我们的新指标理论为Rellich首次观察到的关于具有变化定义域的一参数斯特姆-刘维尔算子族的谱现象提供了新的见解。 显示英文摘要

摘要： The aim of the present manuscript is to develop an index theory for singular Lagrangian systems, with a particular focus on the important class of singular operators given by Bessel type differential operators. The main motivation is to address several challenges posed by singular operators, which appear in a wide range of applications: celestial mechanics (for instance, perturbations in planetary motion), oscillatory systems with time dependent forcing, electromagnetism (such as wave equations in nonuniform media), and quantum mechanics (notably certain Schroedinger equations with periodic potentials). We pursue two principal objectives. First, we establish a spectral flow formula and a Morse Index Theorem for gap-continuous paths of singular Sturm Liouville operators. By means of these index formulas, we construct a Morse index theory for a broad class of Bessel type differential operators and apply it to a family of asymptotic solutions of the gravitational n body problem. Finally, our new index theory provides new insight into a phenomenon first observed by Rellich concerning the spectrum of one-parameter families of Sturm Liouville operators with varying domains.