Geodesic
Inverse Scattering Problem with Underdetermined Data
A. G. Ramm1
1 Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 244-253.

Voir la notice de l'article provenant de la source EDP Sciences

Consider the Schrödinger operator − ∇2 + q with a smooth compactly supported potential q, q = q(x),x ∈ R3. Let A(β,α,k) be the corresponding scattering amplitude, k2 be the energy, α ∈ S2 be the incident direction, β ∈ S2 be the direction of scattered wave, S2 be the unit sphere in R3. Assume that k = k0> 0 is fixed, and α = α0 is fixed. Then the scattering data are A(β) = A(β,α0,k0) = Aq(β) is a function on S2. The following inverse scattering problem is studied: IP: Given an arbitrary f ∈ L2(S2) and an arbitrary small number ϵ> 0, can one find q ∈ C0∞(D) , where D ∈ R3 is an arbitrary fixed domain, such that ||Aq(β) − f(β)|| L2(S2)? A positive answer to this question is given. A method for constructing such a q is proposed. There are infinitely many such q, not necessarily real-valued.
DOI : 10.1051/mmnp/20149516

A. G. Ramm 1

1 Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA 
@article{MMNP_2014_9_5_a15,
     author = {A. G. Ramm},
     title = {Inverse {Scattering} {Problem} with {Underdetermined} {Data}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {244--253},
     publisher = {mathdoc},
     volume = {9},
     number = {5},
     year = {2014},
     doi = {10.1051/mmnp/20149516},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149516/}
}
TY  - JOUR
AU  - A. G. Ramm
TI  - Inverse Scattering Problem with Underdetermined Data
JO  - Mathematical modelling of natural phenomena
PY  - 2014
SP  - 244
EP  - 253
VL  - 9
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149516/
DO  - 10.1051/mmnp/20149516
LA  - en
ID  - MMNP_2014_9_5_a15
ER  - 
%0 Journal Article
%A A. G. Ramm
%T Inverse Scattering Problem with Underdetermined Data
%J Mathematical modelling of natural phenomena
%D 2014
%P 244-253
%V 9
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149516/
%R 10.1051/mmnp/20149516
%G en
%F MMNP_2014_9_5_a15
A. G. Ramm. Inverse Scattering Problem with Underdetermined Data. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 244-253. doi : 10.1051/mmnp/20149516. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149516/

[1] S. Agmon Ann. Scuola Norm. Super. Pisa 1975 151 218

[2] H. Cycon, R. Froese, R. Kirsch, B. Simon. Scrödinger operators. Springer-Verlag, Berlin, 1986.

[3] D. Pearson. Quantum scattering and spectral theory. Acad. Press, London, 1988.

[4] A.G. Ramm Inverse Problems 1988 877 886

[5] A.G. Ramm Milan Journ of Math. 2002 97 161

[6] A.G. Ramm. Inverse problems. Springer, New York, 2005.

[7] A.G. Ramm J.Phys. A, FTC 2010 112001

[8] A.G. Ramm Eurasian Math. Journ (EMJ) 2010 97 111

[9] A.G. Ramm Jour. Stat. Phys. 2007 915 934

[10] A.G. Ramm. Inverse scattering problem with data at fixed energy and fixed incident direction, Nonlinear Analysis: Theory, Methods and Applications. 69 (2008), no. 4, 1478-1484.

[11] A.G. Ramm Afrika Matematika 2011 33 55

[12] A.G. Ramm J. Math. Phys. 2011 123506

[13] A.G. Ramm. Scattering by obstacles. D.Reidel, Dordrecht, 1986.

[14] A.G.Ramm. Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes. Applications to Creating New Engineered Materials. Momentum Press, New York, 2013.

Cité par Sources :