Scattering and inverse scattering for a left-definite Sturm-Liouville problem
(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:
https://lup.lub.lu.se/record/3055579
- author
- Bennewitz, Christer LU ; Brown, B. M. and Weikard, R.
- organization
- publishing date
- 2012
- 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
- 2016-04-01 09:55:57
- date last changed
- 2022-02-09 21:04:57
@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}},
keywords = {{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}},
doi = {{10.1016/j.jde.2012.06.016}},
volume = {{253}},
year = {{2012}},
}