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A generalisation of the inverse spectral theorem of Levitan and Gasymov

Published online by Cambridge University Press:  14 November 2011

Christine Thurlow
Christine Thurlow
Affiliation:
121 The Grove, Ealing, London W5
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Synopsis

In the Inverse Spectral Theorem in the form given by Levitan and Gasymov, necessary and sufficient conditions are given for a non-decreasing function, p(ℷ), to be the spectral function of a Sturm–Liouville problem. In these conditions, p(ℷ) is compared with the spectral function for the particular Strum–Liouville problem

If the method of Levitan and Gasymov's proof is slightly adapted, the necessary and sufficient conditions can be stated in a more general form in which p(ℷ) is compared with the spectral function for any problem of the form

where hO is real and qo(x) locally integrable.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 3-4 , 1979 , pp. 185 - 196
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Gelfand, I. M. and Levitan, B. M.. On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl. 1 (1955), 253–304.Google Scholar
2Levitan, B. M. and Gasymov, M. G.. Determination of a differential equation by two of its spectra. Russian Math. Surveys 19 (1964), 1–63.CrossRefGoogle Scholar
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5Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations, Pt 2 (Oxford Univ. Press, 1958).Google Scholar