具有周期边界条件的两个Sturm-Liouville问题的交叉谱

Intersection of Spectrum for Two Sturm-Liouville Problems with Periodic Boundary Conditions

为研究具有周期边界条件的两个Sturm-Liouville (SL)问题的交叉谱个数,构造一个二维向量SL问题使两个一维SL问题的谱集与该二维向量SL问题的谱集相同,计算出二维SL问题的二重特征值的一个上界MQ,得出二维SL问题的大于MQ的特征值都是单特征值,且只有有限个非单特征值;利用一维SL问题与二维向量SL问题谱集之间的关系,得出具有周期边界条件的两个一维SL问题交叉谱(相同特征值)的个数是有限的,得到具有周期边界条件的一维SL问题的二重特征值个数也是有限的,同时计算出最大二重特征值的上界估计。

In order to study the intersection of the spectra for two Sturm-Liouville(SL) problems with periodic boundary conditions(BCs), a two-dimensional vectorial SL problem is constructed. In this question the spectral sets of two one-dimensional SL problems are equal to the spectral sets of the two-dimensional vectorial SL problem.Then an upper bound MQof the double eigenvalues of the two-dimensional SL problem is calculated. It is concluded that the eigenvalues greater than MQof two-dimensional SL problem are single eigenvalues, and there are only a limited number of non single eigenvalues. Using the relationship between the spectral set of one-dimensional SL problem and two-dimensional vectorial SL problem, it is obtained that the number of cross spectra(same eigenvalues) of two one-dimensional SL problems with periodic BCs is limited, and the number of double eigenvalues of one-dimensional SL problem with periodic BCs is also limited. At the same time, the upper bound estimation of the maximum double eigenvalues is calculated.