Completeness of the system of eigenfunctions of the Sturm–Liouville problem with the singularity

Mathematical modeling of composite materials plays an important role in modern technology, and the solution and study of inverse boundary value problems of heat transfer is impossible without the use of systems of eigenfunctions of the Sturm–Liouville problem for the differential equation with discontinuous coefficients. One of the most important properties of such systems is their completeness in the corresponding spaces. This property of systems allows to prove theorems of existence and uniqueness of both direct problems and inverse boundary value problems of thermal conductivity, and also to prove numerical methods of solving such problems. In this paper, we prove the completeness of the Sturm–Liouville problem in the space $L_2[r_0,r_2]$ for a second-order differential operator with a discontinuous coefficient. This problem arises when investigating and solving the inverse boundary problem of thermal conductivity for a hollow ball consisting of two balls with different temperature conductivity coefficients. Self-conjugacy, injectivity, and positive definiteness of this operator are proved.