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,∞) has been studied in a function-theoretic way by L. de Branges. 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. 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,∞) to a trace-normed system on R, which is in the limit-point case at ±∞. 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