锥中修改的Poisson-Sch积分在无穷远点处的渐近行为及其应用

The stationary Schrödinger equation is an elementary equation in quantum mechanics. It apperars in many important physical modols, for example, nonlinear optics and condensed matter physics. The cone is a special unbounded domain. By constrcting “modified Poisson-Sch kernels” in a cone, we firstly give asymptotic behaviors of “modified Poisson-Sch integrals” at infinity in a cone in this project. Then we consider the applications of asymptotic behaviors of “modified Poisson-Sch integrals” at infinity in a cone: 1. We try to prove integral representations of subsolutions of stationary Schrödinger equation and give asymptotic behaviors at infinity and related properties of exceptional sets of them. 2. We give solutions of Dirichlet-Sch problems. By raising and resolving of this project, we want to generalize classical resunlts about subharmonic functions in a half space. Meanwhile, it plays an important role not only in the self-development of asymptotic behaviors of “modified Poisson-Sch integrals” at infinity in a cone but also further applications of this result.

稳态Schrödinger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。半空间是一个特殊的锥。. 本项目首先通过构造锥中“修改的Poisson-Sch核”，来给出相应的“修改的Poisson-Sch积分”在无穷远点处的渐近行为。 然后，考虑锥中“修改的Poisson-Sch积分”在无穷远点处渐近行为的应用: 1.证明锥中稳态Schrödinger方程弱解的积分表示，并考虑其在无穷远点处的渐近行为以及相关例外集的几何性质;2. 给出锥中Dirichlet-Sch问题的解。. 本项目的提出和解决,不仅推广了半空间中关于次调和函数的经典结果，而且有助于深化对锥中“修改的Poisson-Sch积分”在无穷远点处渐近行为这一结论本身的研究并拓展该结论的进一步应用。

稳态Schrödinger方程是量子力学中的基本方程，其弱解称之为次函数。半空间是一个特殊的锥。本课题系统地构造了锥中“修改的Poisson-Sch核”，同时证明了相应的“修改的Poisson-Sch积分”在无穷远点处的渐近行为。作为应用，解决了下面两个问题：1.证明锥中稳态Schrödinger方程弱解的积分表示，并考虑其在无穷远点处的渐近行为以及相关例外集的几何性质; 2.给出锥中Dirichlet-Sch问题的解。本课题的顺利完成,不仅推广了半空间中关于次调和函数的经典结果，而且有助于深化对锥中“修改的Poisson-Sch积分”在无穷远点处渐近行为这一结论本身的研究并拓展该结论的进一步应用。另外，所得到的结果为研究调和分析与偏微分方程等学科中的相关分析问题提供了新的工作空间和方法。