Abstract: In this paper, we study a class of Sturm-Liouville problems with an infinite number of interior discontinuous points, i.e., Sturm-Liouville problems with an infinite number of discontinuous conditions at interior points. Firstly, we construct a new Hilbert space associated with the discontinuous conditions and define the maximal and minimal operators associated with the discontinuous conditions in the new Hilbert space. And then we discuss the deficiency index of the minimal operator associated with the discontinuous conditions in the new Hilbert space.

Key words: Sturm-Liouville operator, Discontinuity, Discontinuous condition, Deficiency index