边界条件依赖谱参数的非连续Sturm-Liouville算子的谱问题

Spectral distribution problem for Sturm-Liouville with operators the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter

为了丰富Sturm-Liouville(S-L)微分算子的谱理论,研究了闭区间[0,1]上边界条件依赖谱参数的非连续S-L问题。首先利用该问题在直和空间上的等价刻画,给出了非连续S-L问题特征值与连续S-L问题特征值间的交替关系,即在非连续S-L问题的特征值的每个开子区间内都恰有连续S-L问题的一个特征值,进而由连续S-L问题的振荡理论推出非连续S-L问题的振荡理论。然后通过Prüfer变换和Hergloz函数的转换,建立了边界条件依赖谱参数的非连续S-L问题与边界条件为常值的非连续S-L问题的转换,得出转换后的特征值与转换前(除去有限个)的特征值相等。最后通过构造边界条件为常值的非连续S-L问题的特征函数求得其特征值的渐近式,从而得到了边界条件依赖谱参数的非连续S-L问题的特征值的渐近表达式。新的研究方法可推广到对间断点条件依赖谱参数的S-L问题研究。

In order to enrich the spectral theory of Sturm-Liouvillel(S-L)differential operators,the discontinuous S-L problem with boundary conditions dependent on spectral parameters on closed interval[0,1]is studied.Firstly,by using the equivalent characterization of the problem in the direct sum space,the alternating relation between the eigenvalues of the discontinuous S-L problem and the eigenvalues of the continuous S-L problem is given.That is,there is exactly one eigenvalue of the continuous S-L problem in every open subinterval of the eigenvalues of the discontinuous S-L problem,and then the oscillation theory of the discontinuous S-L problem is derived from the oscillation theory of the continuous S-L problem.Through the transformations of Prüfer and Hergloz function,the transformation between the discontinuous S-L problem with boundary conditions dependent spectral parameters and discontinuous S-L problem with constant boundary conditions is established.The obtained converted eigenvalues are equal to those(excluding the finite eigenvalues)before the conversion.Finally,the asymptotic expressions of eigenvalues of discontinuous S-L problems with boundary conditions dependent on spectral parameters are obtained by constructing the eigenfunctions of discontinuous S-L problems with constant boundary conditions.The new research method can be extended to the study of the S-L problem with boundary conditions dependent spectral parameters.