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Inverse Scattering Problem with UnderdeterminedData

Published online by Cambridge University Press:  17 July 2014

A. G. Ramm

Abstract

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Consider the Schrödinger operator −∇2 + q with a smooth compactly supportedpotential 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 thatk =k0> 0 is fixed, andα =α0 is fixed. Then the scattering data areA(β) =A(β,α0,k0)= Aq(β)is a function on S2. The following inverse scatteringproblem 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, suchthat ||Aq(β) −f(β)||L2(S2)<ϵ? A positive answer to this question is given. A method for constructing such aq isproposed. There are infinitely many such q, not necessarily real-valued.

Keywords

underdetermined datainverse scatteringfixed energyfixed incident direction

Information

Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014, pp. 244 - 253
Copyright
© EDP Sciences, 2014

References

Agmon, S.. Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Super. Pisa, 4, (1975), 151-218. Google Scholar
H. Cycon, R. Froese, R. Kirsch, B. Simon. Scrödinger operators. Springer-Verlag, Berlin, 1986.
D. Pearson. Quantum scattering and spectral theory. Acad. Press, London, 1988.
Ramm, A.G.. Recovery of the potential from fixed energy scattering data. Inverse Problems, 4 (1988), 877-886. CrossRefGoogle Scholar
Ramm, A.G.. Stability of solutions to inverse scattering problems with fixed-energy data. Milan Journ of Math., 70 (2002), 97-161. CrossRefGoogle Scholar
A.G. Ramm. Inverse problems. Springer, New York, 2005.
Ramm, A.G.. Uniqueness theorem for inverse scattering problem with non-overdetermined data. J.Phys. A, FTC, 43 (2010), 112001. CrossRefGoogle Scholar
Ramm, A.G.. Uniqueness of the solution to inverse scattering problem with backscattering data. Eurasian Math. Journ (EMJ), 1 (2010), no. 3, 97-111. Google Scholar
Ramm, A.G.. Distribution of particles which produces a “smart” material. Jour. Stat. Phys., 127 (2007), no. 5, 915-934. CrossRefGoogle Scholar
Ramm, A.G.. Inverse scattering problem with data at fixed energy and fixed incident direction, Nonlinear Analysis: Theory, Methods and Applications. 69 (2008), no. 4, 1478-1484. Google Scholar
Ramm, A.G.. Wave scattering by many small bodies and creating materials with a desired refraction coefficient. Afrika Matematika, 22 (2011), no. 1, 33-55. CrossRefGoogle Scholar
Ramm, A.G.. Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave. J. Math. Phys., 52 (2011), 123506. CrossRefGoogle Scholar
A.G. Ramm. Scattering by obstacles. D.Reidel, Dordrecht, 1986.
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.