论文标题
非对称分数椭圆算子的光谱分析
Spectral analysis of a family of nonsymmetric fractional elliptic operators
论文作者
论文摘要
在这项工作中，我们调查了频谱问题$ au =λu$，其中$ a $是涉及左侧和右侧Riemann-Liouville衍生品的分数椭圆操作员。但是，这些操作员是非本地和非对称的，但是，具有某些经典的椭圆特性。特征值对应于某些特殊功能类别的根。与经典的Sturm-Liouville问题相比，最具挑战性的部分是建立分析这些非本地操作员的框架，这需要开发新工具。我们证明了真实特征值的存在，找到所有可能的复杂特征值的范围，探索特征函数的图，并在复杂平面上的特征值分布显示数值发现。
In this work, we investigate the spectral problem $Au = λu$ where $A$ is a fractional elliptic operator involving left- and right-sided Riemann-Liouville derivatives. These operators are nonlocal and nonsymmetric, however, share certain classic elliptic properties. The eigenvalues correspond to the roots of a class of certain special functions. Compared with classic Sturm-Liouville problems, the most challenging part is to set up the framework for analyzing these nonlocal operators, which requires developing new tools. We prove the existence of the real eigenvalues, find the range for all possible complex eigenvalues, explore the graphs of eigenfunctions, and show numerical findings on the distribution of eigenvalues on the complex plane.
