The Backscattering Problem in Three Dimensions
(2001) Abstract
 In this thesis we study the (inverse) backscattering problem for the Schr"odinger operator in $R^3$. We introduce the backscattering transform $B(v)$ of a realvalued potential $vin C_0^infty(R^3)$, and prove that the backscattering 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:
https://lup.lub.lu.se/record/20088
 author
 Lagergren, Robert ^{LU}
 supervisor
 opponent

 Ruiz, Alberto, Universidad Autonoma de Madrid
 organization
 publishing date
 2001
 type
 Thesis
 publication status
 published
 subject
 keywords
 backscattering, 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
 20011210 10:15:00
 ISBN
 9178441602
 language
 English
 LU publication?
 yes
 id
 3ad8a29c4ca24dbeb9923c4b70464167 (old id 20088)
 date added to LUP
 20160401 16:41:58
 date last changed
 20181121 20:43:32
@phdthesis{3ad8a29c4ca24dbeb9923c4b70464167, abstract = {In this thesis we study the (inverse) backscattering problem for the Schr"odinger operator in $R^3$. We introduce the backscattering transform $B(v)$ of a realvalued potential $vin C_0^infty(R^3)$, and prove that the backscattering 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 = {9178441602}, language = {eng}, publisher = {Robert Lagergren, Luzernvägen 12, 352 51 VÄXJÖ,}, school = {Lund University}, title = {The Backscattering Problem in Three Dimensions}, year = {2001}, }