本项目拟研究微分算子的trace公式及其应用。各种各样的trace公式如Selberg trace公式等是泛函分析与其它数学分支之间建立起联系的主要桥梁之一。Krein在上个世纪50年代研究了Hamilton系统-1边值问题的稳定性，发展了trace公式的理论。受此工作启发，本项目拟利用泛函分析的方法研究Hamilton系统Lagrange型边值问题的trace公式，并将之应用到某些具体的物理问题中。对周期类边值问题，Hill-型公式是研究trace公式的出发点。然而，对Hamilton系统Hill-公式的研究均局限于周期类边值问题，其复杂度虽然远高于Hill在1877年的工作，然而结果却十分相似。本项目拟突破周期类边值条件的限制，考虑Lagrange型边值问题的Hill-型公式，其中结果将与周期类边值条件下的结论完全不同。最后，我们希望将trace公式反馈到泛函分析自身的发展中。

In this project, we will mainly consider the trace formula for differential operators and its applications. Various trace formulas, such as Selberg trace formula, play important roles in the applications of functional analysis on the other branches. In 1950's, Krein consider the stability problem of Hamiltonian system with -1 boundary condition, to do this, he developed the theory of trace formula for Hamiltonian system with -1 boundary condition. Inspired by Krein's work, we will consider the Lagrangian boundary problem of Hamiltonian system, moreover, we will use the trace formula to study the stability problem of physics systems. As we done for S-periodic boundary problems of Hamiltonian systems, the Hill-type formula will be the starting point of the study of trace formula. Till now, all the study of Hill-type formula deals with the "periodic"-type boundary problem, although the problem is more complicated, but the results are similar to the original work of Hill. However, for the Lagrange boundary problem, the results will be totally different. Finally, we will use the idea of trace formula for differential operators to study some problem in Functional Analysis, such as the Dixmier trace, Berg-Shaw theorem, ect.