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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References
N.L Achieser and I.M. Glasmann: Theorie der linearen Operatoren im Hilbert Raum. Akademie-Verlag, Berlin, 1954.
N.I. Akhiezer: The Classical Moment Problem. Oliver & Boyd, Edinburgh, 1965.
L. De Branges: Some Hilbert spaces of entire functions. Trans. Amer. Math. Soc. 96 (1960), 259–295; 99 (1961), 118-152; 100 (1960), 73-115; 105 (1962), 43-83.
L. De Branges: Hilbert Spaces of Entire Functions. Prentice Hall, Englewood Cliffs, N.J., 1968.
D.L. Cohn: Measure Theory. Birkhäuser Verlag, Boston, 1980.
H. Dym and A. Iacob: Positive definite extensions, canonical equations and inverse problems. Operator Theory: Advances and Applications, vol.12 Birkhäuser Verlag, Basel, 1984, pp. 141–240.
F.R. Gantmacher: Matrizentheorie. Deutscher Verlag der Wissenschaften, Berlin, 1986.
T. Kato: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, Heidelberg, New York, 1980.
I.S. Kac: Linear relations, generated by a canonical differential equation on an interval with a regular endpoint, and expansibility in eigenfunctions (Russian). Deposited paper 517.9, Odessa, 1984.
M.G. Krein and H. Langer: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume IIK zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77 (1977), 187–236.
M.G. Krein and H. Langer: On some extension problems which are closely connected with the theory of hermitian operators in a space IIK. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part (1): Beiträge zur Analysis 14 (1979), 25–40. Part (2): Beiträge zur Analysis 15 (1981), 27-45.
H. Langer: Spektralfunktionen einer Klasse von Differentialoperatoren zweiter Ordnung mit nichtlinearem Eigenwertparameter. Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 269–301.
H. Langer and H. Winkler: Direct and inverse spectral problems for generalized strings. (submitted).
I.P. Natanson: Theorie der Funktionen einer reellen Veränderlichen. Akademie-Verlag, Berlin, 1981.
A.L. Sakhnovich: Spectral functions of a canonical system of order 2n. Math. USSR Sbornik 71 (1992), 355–369.
L.A. Sakhnovich: The method of operator identities and problems of analysis. Algebra and Analysis 5 (1993), 4–80.
H. Winkler: The inverse spectral problem for canonical systems. Integral Equations Operator Theory 22 (1995), 360–374.
H. Winkler: On transformations of canonical systems. Operator Theory: Advances and Applications, vol. 80, Birkhäuser Verlag, Basel, 1995, pp. 276–288.
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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
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