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

DOI:
https://doi.org/10.1090/S0002-9939-06-08563-7

Published electronically:
October 11, 2006

MathSciNet review:
2262915

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any positive integer and any given distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of 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.

**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**, https://doi.org/10.1006/jmaa.2001.7661**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**

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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:
https://doi.org/10.1090/S0002-9939-06-08563-7

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

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