M. Langer and H. Woracek,
A function space model for canonical systems with an inner singularity,
Acta Sci. Math. (Szeged) 77 (2011), 101–165
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
splitting-and-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.