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Inverse Spectral And Scattering Theory For The Half-Line Left-Definite Sturm-Liouville Problem

Bennewitz, Christer LU ; Brown, B. M. and Weikard, R. (2008) In SIAM Journal on Mathematical Analysis 40(5). p.2105-2131
Abstract
The problem of integrating the Camassa-Holm equation leads to the scattering and inverse scattering problem for the Sturm-Liouville equation -u '' + 1/4 u = lambda wu, where w is a weight function which may change sign but where the left-hand side gives rise to a positive quadratic form so that one is led to a left-definite spectral problem. In this paper the spectral theory and a generalized Fourier transform associated with the equation -u '' + 1/4 u =lambda wu posed on a half-line are investigated. An inverse spectral theorem and an inverse scattering theorem are established. A crucial ingredient of the proofs of these results is a theorem of Paley-Wiener type which is shown to hold true. Additionally, the accumulation properties of... (More)
The problem of integrating the Camassa-Holm equation leads to the scattering and inverse scattering problem for the Sturm-Liouville equation -u '' + 1/4 u = lambda wu, where w is a weight function which may change sign but where the left-hand side gives rise to a positive quadratic form so that one is led to a left-definite spectral problem. In this paper the spectral theory and a generalized Fourier transform associated with the equation -u '' + 1/4 u =lambda wu posed on a half-line are investigated. An inverse spectral theorem and an inverse scattering theorem are established. A crucial ingredient of the proofs of these results is a theorem of Paley-Wiener type which is shown to hold true. Additionally, the accumulation properties of eigenvalues are investigated. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Camassa-Holm equation, Sturm-Liouville, problems, left-definite, inverse scattering problems, inverse spectral problems
in
SIAM Journal on Mathematical Analysis
volume
40
issue
5
pages
2105 - 2131
publisher
SIAM
external identifiers
  • wos:000263103600016
  • scopus:70350100726
ISSN
0036-1410
DOI
10.1137/080724575
language
English
LU publication?
yes
id
3abb56cc-431e-4e1c-9147-4c804fee12c8 (old id 1375646)
date added to LUP
2009-05-08 14:58:47
date last changed
2017-09-03 04:21:05
@article{3abb56cc-431e-4e1c-9147-4c804fee12c8,
  abstract     = {The problem of integrating the Camassa-Holm equation leads to the scattering and inverse scattering problem for the Sturm-Liouville equation -u '' + 1/4 u = lambda wu, where w is a weight function which may change sign but where the left-hand side gives rise to a positive quadratic form so that one is led to a left-definite spectral problem. In this paper the spectral theory and a generalized Fourier transform associated with the equation -u '' + 1/4 u =lambda wu posed on a half-line are investigated. An inverse spectral theorem and an inverse scattering theorem are established. A crucial ingredient of the proofs of these results is a theorem of Paley-Wiener type which is shown to hold true. Additionally, the accumulation properties of eigenvalues are investigated.},
  author       = {Bennewitz, Christer and Brown, B. M. and Weikard, R.},
  issn         = {0036-1410},
  keyword      = {Camassa-Holm equation,Sturm-Liouville,problems,left-definite,inverse scattering problems,inverse spectral problems},
  language     = {eng},
  number       = {5},
  pages        = {2105--2131},
  publisher    = {SIAM},
  series       = {SIAM Journal on Mathematical Analysis},
  title        = {Inverse Spectral And Scattering Theory For The Half-Line Left-Definite Sturm-Liouville Problem},
  url          = {http://dx.doi.org/10.1137/080724575},
  volume       = {40},
  year         = {2008},
}