Indefinite Hamiltonians

Michael Kaltenbäck ()
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Michael Kaltenbäck: TU Wien, Institut für Analysis und Scientific Computing

Chapter 24 in Operator Theory, 2026, pp 613-633 from Springer

Abstract: Abstract It is the aim of the present survey to provide an introduction into the theory of indefinite Hamiltonians and to give an overview over the most important results. Indefinite Hamiltonians can be seen as a distributional generalization of the classical theory of canonical Hamiltonian differential equations as studied among many others by M.G. Kreı̆n and Louis de Branges. The spaces in the background of this theory are no longer Hilbert spaces as in the classical situation, but Pontryagin spaces. This type of spaces can be seen as a Hilbert where the Hilbert space scalar product is replaced by a finite dimensional perturbation. In a similar sense indefinite Hamiltonians can be seen as a certain perturbation of classical Hamiltonians. The theory of indefinite Hamiltonians involves certain reproducing kernel Pontryagin spaces consisting of entire function which constitutes a generalization of the theory of Louis de Branges on Hilbert spaces of entire functions.

Date: 2026
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DOI: 10.1007/978-3-032-16356-1_36

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