The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems

Fleige, Andreas

doi:10.1007/978-3-0348-0667-1_44

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2015

2023

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“…Is there an equally concise analogue of M. G. Krein's solution of the inverse spectral problem? Although there are various results about spectral theory for (1.1) when ω is allowed to be indefinite (we only mention [2,6,8,28,29,47,68] and the references therein), there is still no satisfactory answer to these questions. A first guess could suggest that instead of the class of Stieltjes functions (which are, roughly speaking, determined by not having singularities on the negative real axis) one obtains the entire class of Herglotz-Nevanlinna functions (which may have singularities on the whole real axis).…”

Section: Introductionmentioning

confidence: 99%

The inverse spectral problem for indefinite strings

2015

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Abstract. Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form, υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings.

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