Proof of the deficiency index conjecture

We prove that all possible values of the deficiency indices r,s of a symmetric (formally self-adjoint) differential expression M with complex coefficients which satisfy the well known inequalities are realized. Subject to these inequalities, in 1974 Kogan and Rofe-Beketov showed that all values of r,s which differ by no more than 1 can be realized, in 1978, 1979 Gilbert showed that their difference can be arbitrarily large provided the order of M is large enough. In this paper we show that all values of r and s are realized.