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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
MathSciNet review:
2262915
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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.
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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