Advanced

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

Bennewitz, Christer LU ; Brown, B. M. and Weikard, R. (2012) In Journal of Differential Equations 253(8). p.2380-2419
Abstract
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 =... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Scattering theory, Inverse scattering theory, Left-definite problems, Camassa-Holm equation
in
Journal of Differential Equations
volume
253
issue
8
pages
2380 - 2419
publisher
Elsevier
external identifiers
  • wos:000307604500002
  • scopus:84864309254
ISSN
0022-0396
DOI
10.1016/j.jde.2012.06.016
language
English
LU publication?
yes
id
6b25344d-8573-4c97-903c-469d7227f405 (old id 3055579)
date added to LUP
2012-09-25 14:01:56
date last changed
2017-11-05 03:02:37
@article{6b25344d-8573-4c97-903c-469d7227f405,
  abstract     = {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.},
  author       = {Bennewitz, Christer and Brown, B. M. and Weikard, R.},
  issn         = {0022-0396},
  keyword      = {Scattering theory,Inverse scattering theory,Left-definite problems,Camassa-Holm equation},
  language     = {eng},
  number       = {8},
  pages        = {2380--2419},
  publisher    = {Elsevier},
  series       = {Journal of Differential Equations},
  title        = {Scattering and inverse scattering for a left-definite Sturm-Liouville problem},
  url          = {http://dx.doi.org/10.1016/j.jde.2012.06.016},
  volume       = {253},
  year         = {2012},
}