Characterization of self-adjoint domains for differential operators with interior singular points

We characterize the self-adjoint domains of general even order linear ordinary differential operators which have finite interior singular points in terms of real-parameter solutions of the differential equation. For the purpose we constructed a direct sum space. By the theory of direct sum space and the decomposition of the corresponding maximal domain, we give this complete and analytic characterization in terms of limit-circle solutions. This is for endpoints which are regular or singular and for arbitrary deficiency index.