Singularities of the Schrodinger equation

我讨论了端部非紧流形上薛定谔算子的奇点传播、谱和散射理论。我将欧氏空间上的结果推广到这些流形上，并研究了这种分析本质上需要什么几何结构。我发现端点的存在和它的体积增长率可以分别用无界凸函数的存在和它的凸性强度来表述，并在没有坐标的情况下几何地表示一个模型流形，薛定谔算子理论的方法应用在这个流形上。

I discussed the propagation of singularities and the spectral and scattering theory for the Schrodinger operator on a noncompact manifold with ends. Iextended results on the Euclidean space to those manifolds and studied what geometric structure is essentially needed for such analysis. I found that the existence of an endand its volume growth rate can be rephrased in terms of the existence of an unbounded convex function and its strength of convexity, respectively, and formulated geometrically, without coordinates, one model manifold on which methods of the Schrodinger operator theory applies.