Scattering and inverse scattering for a left-definite Sturm-Liouville problem

Summary, in English

This work develops a scattering and an inverse scattering theory for the Sturm-Liouville equation u '' qu = lambda wu where w may change sign but q >= 0. Thus the left-hand side of the equation gives rise to a positive quadratic form and one is led to a left-definite spectral problem. The crucial ingredient of the approach is a generalized transform built on the Jost solutions of the problem and hence termed the Jost transform and the associated Paley-Wiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the Camassa-Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for -u '' + 1/4 u = lambda wu. (c) 2012 Elsevier Inc. All rights reserved.

• Författare: Christer Bennewitz; B. M. Brown; R. Weikard
• Avdelning/ar: Matematik (naturvetenskapliga fakulteten)
• Publiceringsår: 2012
• Språk: Engelska
• Sidor: 2380-2419
• Publikation/Tidskrift/Serie: Journal of Differential Equations
• Volym: 253
• Issue: 8
• Länkar: http://dx.doi.org/10.1016/j.jde.2012.06.016
• Dokumenttyp: Artikel i tidskrift
• Förlag: Elsevier
• Ämne: Mathematics
• Nyckelord: Scattering theory; Inverse scattering theory; Left-definite problems; Camassa-Holm equation
• Status: Published
• ISBN/ISSN/Övrigt: ISSN: 0022-0396