On the deficiency indices and self-adjointness of symmetric Hamiltonian systems

Abstract

The main purpose of this paper is to investigate the formal deficiency indices N ± of a symmetric first-order system Jf′+Bf=λ Hf 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 ±=0 if I= R resp. N ±=n if I= R + ) and a criterion for their maximality N ±=2n for I= R + (as well as the quasi-regularity). This covers the Kac–Krein and de Branges (Trans. Amer. Math. Soc. 99 (1961) 118) theorems on 2×2 canonical systems and some results from Kogan and Rofe–Beketov (Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75) 5). Some conditions for a canonical system to have intermediate formal deficiency indices 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 S min 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 ∗) . As a byproduct of the regularity result we obtain very short proofs of (generalizations of) the main results of the paper by Kogan and Rofe–Beketov (1974/75).