### Abstract

Original language | English |
---|---|

Pages (from-to) | 101-165 |

Number of pages | 65 |

Journal | Acta Scientiarum Mathematicarum |

Volume | 77 |

Issue number | 1-2 |

Publication status | Published - 2011 |

### Fingerprint

### Keywords

- space model
- canonical systems

### Cite this

*Acta Scientiarum Mathematicarum*,

*77*(1-2), 101-165.

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*Acta Scientiarum Mathematicarum*, vol. 77, no. 1-2, pp. 101-165.

**A function space model for canonical systems.** / Langer, Matthias; Woracek, H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A function space model for canonical systems

AU - Langer, Matthias

AU - Woracek, H.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - space model

KW - canonical systems

UR - http://www.asc.tuwien.ac.at/preprint/2009/asc30x2009.pdf

M3 - Article

VL - 77

SP - 101

EP - 165

JO - Acta Scientiarum Mathematicarum

JF - Acta Scientiarum Mathematicarum

SN - 0001-6969

IS - 1-2

ER -