Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical system Jy(/) = -lHy where the nonnegative Hamiltonian matrix function H(x) is trace-normed on (0,infinity) has been studied in a function-theoretic way by L. de Branges in [5]-[8] We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; cf. [33], [34]. The "formal" defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, infinity) to a trace-normed system on R, which is in the limit-point case at +/-infinity. This allows us to study all possible selfadjoint realizations of the original system by means of a boundary-value problem for the extended canonical system involving an interface condition at 0.