锥中广义调和函数的积分表示

The stationary Schr?dinger equation is an elementary equation in quantum mechanics. It apperars in many important physical modols, for example, nonlinear optics and condensed matter physics. Solutions of this equation are called generalized harmonic functions. The cone is a special unbounded domain. This project takes generalized harmonic functions in a cone as objects of study, by improving the Carleman’s fromula of generalized harmonic functions, constrcting a “modified Green a-function” with respect to the Schr?dinger operator and estimating growth properties at infinity of the “modified Green a-functions” and “modified Poisson a-integrals” sharply, we give integral representations, growth properties at infinity and related properties of exceptional sets of generalized harmonic functions in a cone. By researching these questions, I try to generalize classical resunlts about harmonic functions in a half space(a sepcial cone) in potential theory and further improve elementary theories about generalized harmonic functions in a cone.

稳态Schr?dinger方程是量子力学中的基本方程。它出现在很多重要的物理模型中，例如非线性光学，凝聚态物理等。此方程的解我们称之为广义调和函数。锥是一个特殊的无界区域。本项目以锥中广义调和函数为研究对象，通过证明锥中广义调和函数的Carleman公式，构造一个锥中与稳态Schr?dinger算子相关的“修改的Green a-函数”，并精确估计“修改的Green a-位势”和“修改的Poisson a-积分”在锥中无穷远点处增长性质，来给出锥中广义调和函数的积分表示，无穷远点处的增长性质以及其例外集的相关性质。我试图通过对这些问题的研究推广位势论中关于半空间(一个特殊的锥形区域)中调和函数的经典结果，进一步完善锥中广义调和函数的基本理论。

稳态 Schrödinger 方程是量子力学中的基本方程。 此方程的解我们称之为广义调和函数。锥是一个特殊的无界区域。本项目以锥中广义调和函数为研究对象，通过构造一个锥中与稳态 Schrödinger 算子相关的“修改的 Green a-函数”，得到了以下结果：1. 精确估计“修改的 Green a-位势”和“修改的 Poisson a-积分”在锥中无穷远点处增长性质；2. 给出锥中广义调和函数的积分表示，无穷远点处的增长性质以及其例外集的相关性质。这些问题的解决有助于推广位势论中关于半空间中调和函数的经典结果，进一步完善锥中广义调和函数的基本理论。共发表学术论文9篇, 其中包括6篇SCI论文, 圆满完成了本项目的研究.

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