The inverse spectral problem for indefinite strings

• 成果类型: Article
• 署名作者: Eckhardt, Jonathan; Kostenko, Aleksey
• 署名单位: Cardiff University; University of Vienna
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0629-1
• 发表日期: 2016
• 页码: 939-977
• 关键词: camassa-holm equation; sturm-liouville operators; SHALLOW-WATER EQUATION; ISOSPECTRAL PROBLEM; canonical systems; coefficients; asymptotics; hierarchy; kreins

摘要

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(2) u nu on an interval [0, L), where is a real-valued distribution in H-loc(-1), nu 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.