On Generalized Friedrichs and Krein-von Neumann Extensions and Canonical Systems

Abstract

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so-called Q-function from the Nevanlinna class N. In this note we show to which subclasses Nγ of N the Q-functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q-functions of the generalized Friedrichs and Krein-von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh-Weyl coefficient of the two-dimensional canonical system Jy′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation Tmin with defect numbers (1,1) in the Hilbert space L2H, and Q is equal to the Q-function with respect to the extension corresponding to the boundary condition y1(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q-functions and show under which conditions the generalized Friedrichs or Krein-von Neumann extension exists.