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Inverse spectral theory for Sturm-Liouville problems with finite spectrum

Author(s): Hans Volkmer; Anton Zettl
Journal: Proc. Amer. Math. Soc. 135 (2007), 1129-1132.
MSC (2000): Primary 34B24, 34B09; Secondary 34L05
Posted: October 11, 2006
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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:

1.
F. V. Atkinson, ``Discrete and Continuous Boundary Value Problems'', Academic Press, New York, 1964. MR 0176141 (31:416)

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'', Amer. Math. Soc. Chelsea Publishing, Providence, Rhode Island, (2002). MR 1908601 (2003f:34161)

4.
Q. Kong, H. Wu and A. Zettl, ``Sturm-Liouville Problems with finite spectrum,'' J. Math. Anal. and Appl., (2001), 748-762. MR 1866077 (2002g:34056)

5.
H. Volkmer, ``Eigenvalues associated with Borel sets'', to appear in Real Analysis Exchange.

6.
H. Volkmer, ``Eigenvalue problems of Atkinson, Feller and Krein, and their mutual Relationship'', Electron. J. Differential Equations 2005, No. 48, 15 pp. MR 2135259 (2005k:34103)

7.
A. Zettl, ``Sturm-Liouville Theory'', Mathematical Surveys and Monographs, v. 121, Amer. Math. Soc. 2005. MR 2170950

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Additional Information:

Hans Volkmer
Affiliation: Department of Mathematical Sciences, University of Wisconsin, Milwaukee, Wisconsin 53201
Email: volkmer@csd.uwm.edu

Anton Zettl
Affiliation: Department of Mathematics, Northern Illinois University, De Kalb, Illinois 60115
Email: zettl@math.niu.edu

DOI: 10.1090/S0002-9939-06-08563-7
PII: S 0002-9939(06)08563-7
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
Posted: October 11, 2006
Dedicated: Dedicated to the memory of F.V. Atkinson
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2006, American Mathematical Society

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