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Non-real eigenvalues of singular indefinite Sturm-Liouville operators


Authors: Jussi Behrndt, Qutaibeh Katatbeh and Carsten Trunk
Journal: Proc. Amer. Math. Soc. 137 (2009), 3797-3806
MSC (2000): Primary 47A10; Secondary 47B50
DOI: https://doi.org/10.1090/S0002-9939-09-09964-X
Published electronically: July 10, 2009
MathSciNet review: 2529889
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Abstract: We study a Sturm-Liouville expression with indefinite weight of the form $ \mathrm{sgn}(-d^2/dx^2+V)$ on $ \mathbb{R}$ and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials $ V$ with a certain behaviour at $ \pm\infty$ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to $ -d^2/dx^2+V$ in $ L^2(\mathbb{R})$. The general results are illustrated with examples.


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  • 1. T.Ya. Azizov and I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley and Sons, Chichester, New York, 1989. MR 1033489 (90j:47042)
  • 2. R. Beals, Indefinite Sturm-Liouville problems and half-range completeness, J. Differential Equations 56 (1985), 391-407. MR 780497 (86i:34032)
  • 3. J. Behrndt, On the spectral theory of singular indefinite Sturm-Liouville operators, J. Math. Anal. Appl. 334 (2007), 1439-1449. MR 2338672 (2008e:47104)
  • 4. J. Behrndt, Q. Katatbeh and C. Trunk, Accumulation of complex eigenvalues of indefinite Sturm-Liouville operators, J. Phys. A: Math. Theor. 41 (2008), 244003. MR 2455801
  • 5. J. Behrndt and C. Trunk, On the negative squares of indefinite Sturm-Liouville operators, J. Differential Equations 238 (2007), 491-519. MR 2341434 (2008j:47035)
  • 6. P. Binding and P. Browne, Eigencurves for two-parameter selfadjoint ordinary differential equations of even order, J. Differential Equations 79 (1989), 289-303. MR 1000691 (90e:47038)
  • 7. P. Binding and H. Volkmer, Eigencurves for two-parameter Sturm-Liouville equations, SIAM Review 38 (1996), 27-48. MR 1379040 (97f:34015)
  • 8. J. Bognar, Indefinite Inner Product Spaces, Springer, Berlin, 1974. MR 0467261 (57:7125)
  • 9. B. Ćurgus and H. Langer, A Kreın space approach to symmetric ordinary differential operators with an indefinite weight function, J. Differential Equations 79 (1989), 31-61. MR 997608 (90j:47060)
  • 10. J. Fleckinger and A.B. Mingarelli, On the eigenvalues of nondefinite elliptic operators, Differential equations, Proc. Conf., Birmingham, Ala., 1983, North-Holland Math. Stud. 92, North-Holland, Amsterdam, 1984, 219-227. MR 799351 (86j:35072)
  • 11. W. Greenberg, C.V.M. van der Mee and V. Protopopescu, Boundary Value Problems in Abstract Kinetic Theory, Oper. Theory Adv. Appl.  23, Birkhäuser Verlag, Basel, 1987. MR 896904 (88k:82156)
  • 12. R.L. Hall, Square-well representation for potentials in quantum mechanics, J. Math. Phys. 33 (1992), 3472-3476. MR 1182920 (93g:81018)
  • 13. H.G. Kaper, C.G. Lekkerkerker and J. Hejtmanek, Spectral Methods in Linear Transport Theory, Oper. Theory Adv. Appl. 5, Birkhäuser Verlag, Basel, 1982. MR 685594 (85i:82079)
  • 14. I.M. Karabash and M.M. Malamud, Indefinite Sturm-Liouville operators $ (\mathrm{sgn} x)(-\frac{d^2}{dx^2}+q(x))$ with finite-zone potentials, Operators and Matrices 1 (2007), 301-368. MR 2344680 (2008g:47083)
  • 15. I.M. Karabash, A.S. Kostenko and M.M. Malamud, The similarity problem for $ J$-nonnegative Sturm-Liouville operators, J. Differential Equations 246 (2009), 964-997. MR 2474582
  • 16. I.M. Karabash and C. Trunk, Spectral properties of singular Sturm-Liouville operators with indefinite weight $ \mathrm{sgn} x$, to appear in Proc. Roy. Soc. Edinburgh Sect. A.
  • 17. T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617 (53:11389)
  • 18. I. Knowles, On the number of $ L^{2}$-solutions of second order linear differential equations, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 1-13. MR 529564 (80a:34017)
  • 19. I. Knowles, On the location of eigenvalues of second order linear differential operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 15-22. MR 529565 (80d:34026)
  • 20. Q. Kong, M.  Möller, H. Wu and A. Zettl, Indefinite Sturm-Liouville problems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 3, 639-652. MR 1983691 (2004f:34039)
  • 21. H. Langer, Spectral functions of definitizable operators in Kreın spaces, Proceedings of a Functional Analysis conference held at Dubrovnik, Yugoslavia, November 2-14, 1981, Lecture Notes in Mathematics 948, Springer-Verlag, Berlin-Heidelberg-New York, 1982, 1-46. MR 672791 (84g:47034)
  • 22. B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluwer, Dordrecht, 1991. MR 1136037 (92i:34119)
  • 23. H. Volkmer, Sturm-Liouville problems with indefinite weights and Everitt's inequality, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 1097-1112. MR 1415825 (98a:34096)
  • 24. J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Math. 1258, Springer-Verlag, Berlin, 1987. MR 923320 (89b:47070)
  • 25. J. Weidmann, Lineare Operatoren in Hilberträumen, Teil II, Teubner, Stuttgart, 2003. MR 2382320 (2008k:47002)
  • 26. A. Zettl, Sturm-Liouville Theory, Math. Surveys Monogr. 121, Amer. Math. Soc., Providence, RI, 2005. MR 2170950 (2007a:34005)

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Additional Information

Jussi Behrndt
Affiliation: Department of Mathematics MA 6–4, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: behrndt@math.tu-berlin.de

Qutaibeh Katatbeh
Affiliation: Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan
Email: qutaibeh@yahoo.com

Carsten Trunk
Affiliation: Department of Mathematics, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
Email: carsten.trunk@tu-ilmenau.de

DOI: https://doi.org/10.1090/S0002-9939-09-09964-X
Received by editor(s): November 14, 2008
Received by editor(s) in revised form: February 14, 2009, and February 23, 2009
Published electronically: July 10, 2009
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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