非连续Sturm-Liouville算子的谱分布及其逆特征值问题

The spectral distribution and the inverse eigenvalue problem for the discontinuous Sturm-Liouville operators

研究了定义在[0,1]区间且在点t0∈(0,1)具有界面条件的Sturm-Liouville算子的特征值与定义在子区间[0,t0]与[t0,1]上的两个Sturum-Liouville算子的特征值分布及其逆特征值问题.利用Weyl-Titchmarsh-m-函数的单调性态,证明了这三组谱之间具有交错性关系,并证明了若子区间上的两组谱不相交,则可由这三组谱唯一确定势函数q(x)与边值条件中的参数h和H.

The eigenvalue and the inverse eigenvalue problems of the Sturm-Liouville operators defined respectively on[0,1][0,t^0 ]and [t^0 ,1]are considered.By using the monotonicity of the Weyl-Titchmarsh-m-function,it is shown that the three spectra are alternate,and the potential q （x）and the parameters h,H in the boundary conditions can be uniquely determined by the three spectra if the spectra of the operators defined on subintervals are disjoint.