Boundary-value problems for two-dimensional canonical systems

Abstract

The two-dimensional canonical systemJy′=−lHy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ∞) 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; of. [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, ∞) to a trace-normed system onℜ 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 boundaryvalue problem for the extended canonical system involving an interface condition at 0.