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

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Author

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.

Publishing year

2012

Language

English

Pages

2380-2419

Publication/Series

Journal of Differential Equations

Volume

253

Issue

8

Document type

Journal article

Publisher

Elsevier

Topic

  • Mathematics

Keywords

  • Scattering theory
  • Inverse scattering theory
  • Left-definite problems
  • Camassa-Holm equation

Status

Published

ISBN/ISSN/Other

  • ISSN: 0022-0396