Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Inverse spectral theory for Sturm-Liouville problems with finite spectrum

Authors: Hans Volkmer and Anton Zettl
Journal: Proc. Amer. Math. Soc. 135 (2007), 1129-1132
MSC (2000): Primary 34B24, 34B09; Secondary 34L05
Published electronically: October 11, 2006
MathSciNet review: 2262915
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any positive integer $ n$ and any given $ n$ distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of $ n$ numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

References [Enhancements On Off] (What's this?)

  • 1. F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
  • 2. P. Binding and H. Volkmer, ``Prüfer angle asymptotics for Atkinson's semi-definite Sturm-Liouville eigenvalue problem'', to appear in Math. Nachr.
  • 3. F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601
  • 4. Q. Kong, H. Wu, and A. Zettl, Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl. 263 (2001), no. 2, 748–762. MR 1866077,
  • 5. H. Volkmer, ``Eigenvalues associated with Borel sets'', to appear in Real Analysis Exchange.
  • 6. Hans Volkmer, Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relationship, Electron. J. Differential Equations (2005), No. 48, 15. MR 2135259
  • 7. Anton Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. MR 2170950

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34B24, 34B09, 34L05

Retrieve articles in all journals with MSC (2000): 34B24, 34B09, 34L05

Additional Information

Hans Volkmer
Affiliation: Department of Mathematical Sciences, University of Wisconsin, Milwaukee, Wisconsin 53201

Anton Zettl
Affiliation: Department of Mathematics, Northern Illinois University, De Kalb, Illinois 60115

Keywords: Sturm-Liouville problems, finite spectrum, inverse spectral theory
Received by editor(s): March 21, 2005
Received by editor(s) in revised form: November 11, 2005
Published electronically: October 11, 2006
Dedicated: Dedicated to the memory of F.V. Atkinson
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society