The inverse spectral problem for indefinite strings

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

27 pages

References in corpus (5)

• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• Canonical systems and de Branges spaces
• On the isospectral problem of the dispersionless Camassa-Holm equation
• Spectral asymptotics for canonical systems
• Quadratic operator pencils associated with the conservative Camassa-Holm flow

Cited by in corpus (17)

• The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
• Spectral asymptotics for canonical systems
• Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
• A view of the peakon world through the lens of approximation theory
• The Classical Moment Problem and Generalized Indefinite Strings
• The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
• On the absolutely continuous spectrum of generalized indefinite strings
• Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
• A Lagrangian view on complete integrability of the two-component Camassa-Holm system
• Unique solvability of a coupling problem for entire functions
• On a class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl--Titchmarsh theory
• Trace formulas and inverse spectral theory for generalized indefinite strings
• Continued fraction expansions of Herglotz-Nevanlinna functions and generalized indefinite strings of Stieltjes type
• On the absolutely continuous spectrum of generalized indefinite strings II
• Generalized indefinite strings with purely discrete spectrum
• On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations
• The conservative Camassa-Holm flow with step-like irregular initial data