The Back-scattering Problem in Three Dimensions
(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:
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
- 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 10:15:00
- ISBN
- 91-7844-160-2
- language
- English
- LU publication?
- yes
- id
- 3ad8a29c-4ca2-4dbe-b992-3c4b70464167 (old id 20088)
- date added to LUP
- 2016-04-01 16:41:58
- date last changed
- 2018-11-21 20:43:32
@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}}, keywords = {{back-scattering; Schrödinger operator; inverse scattering; Mathematics; Matematik}}, language = {{eng}}, publisher = {{Robert Lagergren, Luzernvägen 12, 352 51 VÄXJÖ,}}, school = {{Lund University}}, title = {{The Back-scattering Problem in Three Dimensions}}, year = {{2001}}, }