斜对角无穷维Hamilton算子的点谱和特征函数系辛结构的非退化性

On the Point Spectrum and Non-Degenerate Symplectic Structure of Eigenfunction Systems of Off-Diagonal Infinite Dimensional Hamiltonian Operators

中文关键词:  点谱  非退化性  特征函数系  斜对角无穷维Hamilton算子

英文关键词:point spectrum  non-degenerate symplectic structure  eigenfunction system  off-diagonal infinite dimensional Hamiltonian operator

中文摘要:

      本文研究斜对角无穷维Hamilton算子$H=\begin{pmatrix}0&B\\C&0\end{pmatrix}$的点谱和特征函数系辛结构的非退化性, 给出斜对角无穷维Hamilton算子$H$的特征函数系具有非退化辛结构的充分必要条件. 基于此, 进一步刻画了斜对角无穷维Hamilton算子$H$的点谱分别包含于实轴、虚轴以及其它区域的充分必要条件. 最后, 以板弯曲问题和弦振动问题中导出的斜对角无穷维Hamilton算子为例, 验证了所得结论的正确性.

英文摘要:

      The point spectrum and non-degenerate symplectic structure of eigenfunction systems of off-diagonal infinite dimensional Hamiltonian operator $H=\big(\begin{smallmatrix}0& B\\ C& 0\end{smallmatrix}\big)$ are studied in this article. The necessary and sufficient conditions for the eigenfunction systems of off-diagonal infinite dimensional Hamiltonian operator $H$ to have non-degenerate symplectic structure are given. Further, the necessary and sufficient conditions for point spectrum to be contained in real axis, imaginary axis and other areas are obtained for off-diagonal infinite dimensional Hamiltonian operator $H$, respectively. As an illustrating example, off-diagonal infinite dimensional Hamiltonian operators derived from the plate bending problem and string vibration problem are used to justify the conclusions.