锥中次函数的Riesz-Herglotz表示定理及其应用

稳态Schrodinger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。 本项目首先研究锥中次函数的Riesz-Herglotz表示定理。通过证明"截断锥"中的"Green公式"，估计(与稳态Schrodinger算子相关的)"Green位势"和"Poisson积分"在锥中无穷远点处增长性质，来给出锥中次函数的Riesz-Herglotz表示定理，无穷远点处的增长性质以及其例外集的相关性质。 然后考虑锥中次函数Riesz-Herglotz表示定理的应用: 1.证明锥中次函数的Phragmén-Lindelof型定理;2. 给出锥中（与稳态Schrodinger算子相关的）Dirichlet问题解的通式。 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次函数Riesz-Herglotz表示定理本身的研究并拓展该定理的进一步应用。

The stationary Schrodinger equation is an elementary equation in quantum mechanics, subsoultions of this equation are called subfunctions. The half space is a special cone. In this project we first want to research: The Riesz-Herglotz representation theorem of subfunctions in a cone. By proving the "Green formula" in a "truncated" cone, estimating growth properties at infinity of (with respect to the Schrodinger operator) "Green potentials" and "Poisson integrals", we give the Riesz-Herglotz representation theorem, growth properties at infinity and related properties of exceptional sets of them. As applications of the Riesz-Herglotz representation theorem of subfunctions in a cone, we try to prove Phragmén-Lindelof type theorems of subfunctions and give solutions of (with respect to the Schrodinger operator) Dirichlet problems in a cone. 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 the Riesz-Herglotz representation theorem of subfunctions in a cone but also further applications of this theorem.