Singular Sturm-Liouville Operators With Transmission Conditions

In this paper,we study a class of singular Sturm-Liouville(S-L) operators with transmission conditions. In order to study the self-adjointness of these singular S-L operators, firstly, we introduced the self-adjointness of regular S-L operators with transmission conditions, including the operators with one discontinuous point or finite discon-tinuous points. On this basis, applying the classical theory of Weyl circle sets Cb, we give the equation, center and radius of the Weyl circle and establish the necessary and sufficient conditions for the function m which in the Weyl circle, and obtained the set of Weyl circle. We give the relationship between the number of square-integrable solutions in H and the limit of Cb which is a circle or a point when b→∞. We give the definition of the limit-point and limit-circle of the singular S-L operators with transmission conditions, and give equivalence relations of the deficiency index, limit-point (limit-circle), and the number of the square-integrable solutions in H. In order to study the self-adjointness of these operators, we give the definition of the maximal and minimal domains associated with transmission conditions. Combining of the well-known GKN theorem, we give analytic description of self-adjoin domain of singular Sturm-Liouville operator with transmission condi-tions in the cases of the limit-circle(limit-point) respectively.