具有无穷多个不连续点Sturm-Liouville算子的Friedrichs扩张

The Friedridis Extension of Sturm-Liouville Operators with an Infinite Number of Interior Discontinuous Points

研究了一类内部具有无穷多个不连续点的Sturm-Liouville问题,即内部具有无穷多个转移条件的Sturm-Liouville问题.把此类问题放到一个新的空间中去考虑,定义了与转移条件相关联的最小算子Cmin和最大算子Cmax,给出了最小算子Cmin是下有界的一个充分条件,进一步由边界条件刻画了具有下有界的最小算子Cmin的Friedrichs扩张.

In this paper,a class of Sturm-Liouville problems with an infinite number of interior discontinuous points,i.e.,Sturm-Liouville problems with an infinite number of transmission conditions at interior points,are studied.Firstly,we construct a new Hilbert space associated with the transmission conditions and define the maximal and minimal operators Cmax,Cmin associated with the transmission conditions in the new Hilbert space,and give some properties of the operators Cmax,Cmin.And then we obtain a sufficient condition for the existence of lower bound for the operator Cmin and further characterize the Friedridis extension.