ACTA issues

Pontryagin spaces of entire functions. V

Michael Kaltenbäck, Harald Woracek

Acta Sci. Math. (Szeged) 77:1-2(2011), 223-336

Abstract. The spectral theory of a two-dimensional canonical (or `Hamiltonian') system is closely related with two notions, depending whether Weyl's limit circle or limit point case prevails. Namely, with its monodromy matrix or its Weyl coefficient, respectively. A Fourier transform exists which relates the differential operator induced by the canonical system to the operator of multiplication by the independent variable in a reproducing kernel space of entire $2$-vector valued functions or in a weighted $L^2$-space of scalar valued functions, respectively. Motivated from the study of canonical systems or Sturm--Liouville equations with a singular potential and from other developments in Pontryagin space theory, we have suggested a generalization of canonical systems to an indefinite setting which includes a finite number of inner singularities. We have constructed an operator model for such `indefinite canonical systems'. The present paper is devoted to the construction of the corresponding monodromy matrix or Weyl coefficient, respectively, and of the Fourier transform.

AMS Subject Classification (1991): 47E05, 46C20, 47B25, 46E22

Keyword(s): canonical system, Pontryagin space boundary triple, maximal chain of matrices, Weyl coefficient

Received August 4, 2009, and in revised form January 27, 2011. (Registered under 90/2009.)