On the number of square integrable solutions and self–adjointness of symmetric first order systems of differential equations

The main purpose of this paper is to investigate the formal deficiency indices N±(I) of a symmetric first order system Jf ′ +Bf = λH f on an interval I, where I = R or I = R±. Here J,B,H are n × n matrix valued functions and the Hamiltonian H ≥ 0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers N±(R) = 0 (resp. N±(R±) = n) and a criterion for their maximality N±(R+) = 2n. Some conditions for a canonical system to have intermediate numbers N±(R+) are presented, too. We also obtain a generalization of the well–known Titchmarsh–Sears theorem for second order Sturm– Liouville type equations. This contains results due to Lidskii and Krein as special cases. We present two approaches to the above problems: one dealing with formal deficiency indices and one dealing with (ordinary) deficiency indices. Our main (non– formal) approach is based on the investigation of a symmetric linear relation Smin which is naturally associated to a first order system. This approach works in the framework of extension theory and therefore we investigate in detail the domain D(S min) of S min. In particular, we prove the so called regularity theorem for D(S ∗ min). The regularity result allows us to construct a bridge between the ”formal” and ”non–formal” approaches by establishing a connection between the formal deficiency indices N± and the usual deficiency indicesN±(Smin). In particular we have N± = N± for definite systems. As a byproduct of the the regularity result we obtain very short proofs of (generalizations of) the main results of the paper by Kogan and Rofe–Beketov [18] as well as a criterion for the quasi–regularity of canonical systems. This covers the Kac–Krein theorem and some results from [18].