Pontryagin spaces of entire functions. VI

Abstract

Recently, a generalization to the Pontryagin space setting of the notion of canonical (Hamiltonian) systems which involves a finite number of inner singularities has been given. The spectral theory of indefinite canonical systems was investigated with help of an operator model. This model consists of a Pontryagin space boundary triple and was constructed in an abstract way. Moreover, the construction of this operator model involves a procedure of splitting-and-pasting which is technical but at the present stage of development in general inevitable.In this paper we provide an isomorphic form of this operator model which acts in a finite dimensional extension of a function space naturally associated with the given indefinite canonical system. We give explicit formulae for the model operator and the boundary relation. Moreover, we show that under certain asymptotic hypotheses the procedure of splittingand-pasting can be avoided by employing a limiting process.We restrict attention to the case of one singularity. This is the core of the theory, and by making this restriction we can significantly reduce the technical effort without losing sight of the essential ideas.

A local inverse spectral theorem for Hamiltonian systems

“…A local uniqueness theorem analogous to theorem 1.2 remains true in the setting of indefinite Hamiltonian systems as introduced and studied in [46][47][48] 4 . The proof is word by word the same, only at some places one has to refer to Pontryagin space theory instead of classical Hilbert space results.…”