The inverse spectral problem for indefinite strings

TL;DR

This paper develops spectral theory for a new class of indefinite strings related to nonlinear wave equations, providing an inverse spectral solution analogous to Krein's classical results for vibrating strings.

Contribution

It introduces generalized indefinite strings with distributional coefficients and solves the inverse spectral problem for them, extending Krein's classical theory.

Findings

Established spectral theory for indefinite strings with distributional coefficients.

Provided an inverse spectral solution analogous to Krein's for these strings.

Extended classical vibrating string theory to more general, indefinite cases.

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 \[-u"=z\,u\,\omega+z^2u\,\upsilon\] on an interval $[0,L)$, where $\omega$ is a real-valued distribution in $H^{-1}_{\mathrm{loc}}[0,L)$, $\upsilon$ 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.