A class of Nevanlinna functions related to singular Sturm-Liouville problems
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- by Seppo Hassi, Manfred Möller and Henk de Snoo PDF
- Proc. Amer. Math. Soc. 134 (2006), 2885-2893 Request permission
Abstract:
The class of Nevanlinna functions consists of functions which are holomorphic off the real axis, which are symmetric with respect to the real axis, and whose imaginary part is nonnegative in the upper halfplane. The Kac subclass of Nevanlinna functions is defined by an integrability condition on the imaginary part. In this note a further subclass of these Kac functions is introduced. It involves an integrability condition on the modulus of the Nevanlinna functions (instead of the imaginary part). The characteristic properties of this class are investigated. The definition of the new class is motivated by the fact that the Titchmarsh-Weyl coefficients of various classes of Sturm-Liouville problems (under mild conditions on the coefficients) actually belong to this class.References
- Louis de Branges, Some Hilbert spaces of entire functions, Trans. Amer. Math. Soc. 96 (1960), 259–295. MR 133455, DOI 10.1090/S0002-9947-1960-0133455-X
- Louis de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152. MR 133456, DOI 10.1090/S0002-9947-1961-0133456-2
- Louis de Branges, Some Hilbert spaces of entire functions. III, Trans. Amer. Math. Soc. 100 (1961), 73–115. MR 133457, DOI 10.1090/S0002-9947-1961-0133457-4
- Louis de Branges, Some Hilbert spaces of entire functions. IV, Trans. Amer. Math. Soc. 105 (1962), 43–83. MR 143016, DOI 10.1090/S0002-9947-1962-0143016-6
- William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556, DOI 10.1007/978-3-642-65755-9
- Seppo Hassi, Michael Kaltenbäck, and Henk de Snoo, Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass $\textbf {N}_1$ of Nevanlinna functions, J. Operator Theory 37 (1997), no. 1, 155–181. MR 1438205
- Seppo Hassi, Heinz Langer, and Henk de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers $(1,1)$, Topics in operator theory, operator algebras and applications (Timişoara, 1994) Rom. Acad., Bucharest, 1995, pp. 115–145. MR 1421120
- S. Hassi, M. Möller, and H.S.V. de Snoo, “Differential operators and spectral functions”, Proceedings of UMC-2001, 11, 313–322 (2002).
- Seppo Hassi, Manfred Möller, and Henk de Snoo, Sturm-Liouville operators and their spectral functions, J. Math. Anal. Appl. 282 (2003), no. 2, 584–602. MR 1989113, DOI 10.1016/S0022-247X(03)00189-6
- Seppo Hassi, Manfred Möller, and Henk de Snoo, Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter, J. Math. Anal. Appl. 295 (2004), no. 1, 258–275. MR 2064424, DOI 10.1016/j.jmaa.2004.03.042
- I. S. Kac, On integral representations of analytic functions mapping the upper half-plane onto a part of itself, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 3(69), 139–144 (Russian). MR 0080745
- American Mathematical Society Translations, Series 2. Vol. 103: Nine papers in analysis, American Mathematical Society, Providence, R.I., 1974. MR 0328627
- H. Winkler, The inverse spectral problem for canonical systems, Integral Equations Operator Theory 22 (1995), no. 3, 360–374. MR 1337383, DOI 10.1007/BF01378784
Additional Information
- Seppo Hassi
- Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland
- Email: sha@uwasa.fi
- Manfred Möller
- Affiliation: Department of Mathematics, University of the Witwatersrand, Wits, 2050, South Africa
- MR Author ID: 212175
- Email: manfred@maths.wits.ac.za
- Henk de Snoo
- Affiliation: Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, Nederland
- Email: desnoo@math.rug.nl
- Received by editor(s): August 3, 2004
- Received by editor(s) in revised form: April 12, 2005
- Published electronically: April 7, 2006
- Additional Notes: The authors gratefully acknowledge the support of the NRF of South Africa under GUN 2053746, of the Research Institute for Technology of the University of Vaasa, and of the Dutch Association for Mathematical Physics under MF04/62a.
- Communicated by: Joseph A. Ball
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2885-2893
- MSC (2000): Primary 30D15; Secondary 34B20
- DOI: https://doi.org/10.1090/S0002-9939-06-08295-5
- MathSciNet review: 2231612