特征算子谱表示与特征展开的研究

Study on spectral representation and eigenexpansion based on eigen-operators

深入研究了特征算子的谱表示与特征展开。给出了特征微分方程格林函数与厄尔密特微分算子及厄尔密特积分算子的关系式,以及厄尔密特微分算子与厄尔密特积分算子两者互逆的关系式;给出了厄尔密特微分算子的谱表示,指出有限区间斯?刘特征方程不能用于实现无穷维的谱表示式,厄尔密特微分算子的谱表示比诺伊曼研究简单清楚得多,具有优越性;给出了厄尔密特积分算子的特征展开(特征分解),具有理论一般性与全面性的优点,对文献[2]中将其用于研究特征谱表示的不正确论述进行了更正;给出了最优特征展开中长球面波函数命名的物理与几何意义。

The spectral representation and expansion based on eigen-operators were deeply studied.The relations between Green’s function of characteristic differential equation and Hermitian differential and integral operators were given.The inverse relations between Hermitian differential operator and Hermitian integral operator were also studied.The spectral representation of Hermitian differential operators was given.It was also shown that the S-L eigen-equation cannot be used to realize the spectral representations of infinite dimensions in a finite interval.The method is much simpler and clearer than that of Neumann,and has advantages.The eigen-expansion(eigen-decomposition)of the Hermitian integral operator was given,which has the advantages of theoretical generality and comprehensiveness.The incorrect discussion in Wang et al[2]was correct that it is used to study the representation of characteristic spectrum.The physical and geometric meanings of naming for the long spherical wave function in optimal eigen-expansion were given.