M. Kaltenbäck, H. Woracek:

"Pontryagin spaces of entire functions VI";

in: "ASC Report 22/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

In the theory of two-dimensional canonical (also called `Hamiltonian´)

systems, the notion of the Titchmarsh-Weyl coefficient associated to a

Hamiltonian function plays a vital role. A cornerstone in the spectral

theory of canonical systems is the Inverse Spectral Theorem due to Louis

de Branges which states that the Hamiltonian function of a given system

is (up to changes of scale) fully determined by its Titchmarsh-Weyl co-

efficient. Much (but not all) of this theory can be viewed and explained

using the theory of entire operators due to Mark G.Kre˘ın.

Motivated from the study of canonical systems or Sturm-Liouville

equations with a singular potential, and from other developments in the

indefinite world, it was a long standing open problem to find an indefinite

(Pontryagin space) analogue of the notion of canonical systems, and to

prove a corresponding analogue of de Branges´ Inverse Spectral Theorem.

We gave a definition of an indefinite analogue of a Hamiltonian function

and elaborated the operator theory of such `indefinite canoncial systems´

in previous work. In the present paper we prove the corresponding version

of the Inverse Spectral Theorem.

canonical system, Pontryagin space, Inverse Spectral Theorem

http://www.asc.tuwien.ac.at/preprint/2009/asc22x2009.pdf

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