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The Back-scattering Problem in Three Dimensions

Lagergren, Robert LU (2001)
Abstract
In this thesis we study the (inverse) back-scattering problem for the Schr"odinger operator in $R^3$. We introduce the back-scattering transform $B(v)$ of a real-valued potential $vin C_0^infty(R^3)$, and prove that the back-scattering data associated to $v$ determine $B(v)$. Under the assumption that the Schr"odinger operator $H_v=-Delta +v$ has no eigenvectors in $L^2(R^3)$ it is shown that $B(v)$ may be expressed in terms of the wave group $K_v(t)=sin(tsqrt{H_v})/sqrt{H_v}$. We prove also that the mapping $vmapsto B(v)$ is a homeomorphism in a neighbourhood of the origin in the Banach space $X_0^r$, which is the completion of $C_0^infty(R^3;R)$ w.r.t. the norm $fmapstosum_{|a|=1}|d^af|_{L^1}$.
Please use this url to cite or link to this publication:
author
opponent
  • Ruiz, Alberto, Universidad Autonoma de Madrid
organization
publishing date
type
Thesis
publication status
published
subject
keywords
back-scattering, Schrödinger operator, inverse scattering, Mathematics, Matematik
pages
67 pages
publisher
Robert Lagergren, Luzernvägen 12, 352 51 VÄXJÖ,
defense location
Matematikcentrum, Sölvegatan 18, Sal MH:C
defense date
2001-12-10 10:15
ISSN
1404-0034
ISBN
91-7844-160-2
language
English
LU publication?
yes
id
3ad8a29c-4ca2-4dbe-b992-3c4b70464167 (old id 20088)
date added to LUP
2007-05-28 08:39:01
date last changed
2016-09-19 08:44:57
@phdthesis{3ad8a29c-4ca2-4dbe-b992-3c4b70464167,
  abstract     = {In this thesis we study the (inverse) back-scattering problem for the Schr"odinger operator in $R^3$. We introduce the back-scattering transform $B(v)$ of a real-valued potential $vin C_0^infty(R^3)$, and prove that the back-scattering data associated to $v$ determine $B(v)$. Under the assumption that the Schr"odinger operator $H_v=-Delta +v$ has no eigenvectors in $L^2(R^3)$ it is shown that $B(v)$ may be expressed in terms of the wave group $K_v(t)=sin(tsqrt{H_v})/sqrt{H_v}$. We prove also that the mapping $vmapsto B(v)$ is a homeomorphism in a neighbourhood of the origin in the Banach space $X_0^r$, which is the completion of $C_0^infty(R^3;R)$ w.r.t. the norm $fmapstosum_{|a|=1}|d^af|_{L^1}$.},
  author       = {Lagergren, Robert},
  isbn         = {91-7844-160-2},
  issn         = {1404-0034},
  keyword      = {back-scattering,Schrödinger operator,inverse scattering,Mathematics,Matematik},
  language     = {eng},
  pages        = {67},
  publisher    = {Robert Lagergren, Luzernvägen 12, 352 51 VÄXJÖ,},
  school       = {Lund University},
  title        = {The Back-scattering Problem in Three Dimensions},
  year         = {2001},
}