一类Schrodinger算子的离散谱的渐近行为

This project is devoted to asymptotic behaviors of the discrete spectrums of Schrodinger operators, including asymptotic behaviors of the discrete spectrums of Schrodinger operators with real-valued potentials, the resolvent expansion and asymptotic behaviors of the discrtet spectrums of Schrodinger operators with complex-valued potentials. We first study the resolvent expansion of Schrodinger operators. Then, we construct the Birman-Schwinger kernel and study its asymptotic expansion. Next, we get the asymptotic expansions of the spectrums of Birman-Schwinger kernel by using the functional analysis and the expansion of Schrodinger operators and the spetcrums of Birman-Schwinger kernel, we can get the asymptotic behaviors of the discrete spectrums of Schrodinger operators. The main aim of this project is to exploare the properties of Schrodinger operators with complex-valued potentials, to devolop the theory of Schrodinger operators with complex-valued critical potentials, to open out the asymptotic behaviors of the discrete eigenvalues of Schrodinger operators with critical potentials.

本项目研究Schrodinger算子的离散谱的渐近行为包括带实值临界势能的Schrodinger算子的离散谱的渐近行为和带复值临界势能的Schrodinger算子的预解式展式、离散谱的渐近行为。首先研究Schrodinger算子的预解式展式，然后构造Birman-Schwinger核算子，并得到Birman-Schwinger算子的渐近展式。其次用谱分析的方法和Birman-Schwinger和算子的渐近展式得到Birman-Schwinger算子的谱的渐近展式，最后根据Birman-Schwinger核算子和Schrodinger算子的谱的关系得到Schrodinger算子的渐近行为。本项目旨在探索复值临界势能Schrodinger算子的性质，发展带临界势能的Schrodinger算子的理论，解释带临界势能的Schrodinger算子的离散谱的渐近行为。

本项目研究了一类带有临界势能的Schrodinger算子的离散谱的渐近行为，以及Schrodinger算子的一些应用。首先研究了Schrodinger算子的预解式的渐近展式，然后得到Schrodinger算子所对应的Birman-Schwinger核算子的渐近展式。其次再利用谱分析方法和Birman-Schwinger核算子的渐近展式得到Birman-Schwinger核算子的离散谱的渐近展式，最后根据两类算子离散谱的对应关系，得到Schrodinger算子的谱的渐近行为。另外，我们还研究了Schrodinger算子相关的一些微分方程的问题，研究了随机偏微分方程的随机吸引子的存在性和连续性，研究了一些非线性微分方程解的存在性，得到了一些非线性偏微分方程的精确解析解。