Abstract
We consider a singular two-dimensional canonical system Jy’ = −zHy on [0, L) such that at L Weyl’s limit point case holds. Here H is a real and nonnegative definite matrix function, the so-called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems (or their Hamiltonians H) and their Titchmarsh-Weyl coefficients is a bijection between the class of trace normed Hamiltonians H and the class of Nevanlinna functions. In this note we show that the Hamiltonian H of a canonical system with a semibounded spectrum has the property det H = 0 and that its components are functions of locally bounded variation. Further, a characterization of the class of Hamiltonians corresponding to canonical systems with a finite number of negative (or positive) eigenvalues is given.
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Dedicated to Heinz Langer on the occasion of his 60th birthday
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Winkler, H. (1998). Canonical systems with a semibounded spectrum. In: Dijksma, A., Gohberg, I., Kaashoek, M.A., Mennicken, R. (eds) Contributions to Operator Theory in Spaces with an Indefinite Metric. Operator Theory Advances and Applications, vol 106. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8812-7_22
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DOI: https://doi.org/10.1007/978-3-0348-8812-7_22
Publisher Name: Birkhäuser, Basel
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