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November  2010, 4(4): 599-618. doi: 10.3934/ipi.2010.4.599

The quadratic contribution to the backscattering transform in the rotation invariant case

Ingrid Beltiţă 1, and  Anders Melin 2,

1. 

Institute of Mathematics of the Romanian Academy, Bucharest, PO Box 1–764, Romania
2. 

Lund University, Box 118, S-22100, Lund, Sweden

Received  December 2008 Published  September 2010

Considerations of the backscattering data for the Schrödinger operator $H_v= -\Delta+ v$ in $\RR^n$, where $n\ge 3$ is odd, give rise to an entire analytic mapping from $C_0^\infty ( \RRn)$ to $C^\infty (\RRn)$, the backscattering transformation. The aim of this paper is to give formulas for $B_2(v, w)$ where $B_2$ is the symmetric bilinear operator that corresponds to the quadratic part of the backscattering transformation and $v$ and $w$ are rotation invariant.
Keywords: spherical averages., backscattering transformation, Born approximation.
Mathematics Subject Classification: Primary: 81U025; Secondary: 35R3.
Citation: Ingrid Beltiţă, Anders Melin. The quadratic contribution to the backscattering transform in the rotation invariant case. Inverse Problems and Imaging, 2010, 4 (4) : 599-618. doi: 10.3934/ipi.2010.4.599
References:
[1]

I. Beltiţă and A. Melin, Analysis of the quadratic term in the backscattering transformation, Math. Scand., 105 (2009), 218-234.

[2]

I. Beltiţă and A. Melin, Local smoothing for the backscattering transformation, Comm. Partial Diff. Equations, 34 (2009), 233-256.

[3]

L. Hörmander, "The Analysis of Linear Partial Differential Operators'' (I-IV), Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1983-1985.

[4]

A. Melin, Smoothness of higher order terms in backscattering, in "Wave Phenomena and Asymptotic Analysis,'' RIMS Kokyuroku, 1315 (2003), 43-51.

[5]

A. Melin, Some transforms in potential scattering in odd dimension, in "Inverse Problems and Spectral Theory,'' Contemp. Math., 348, Amer. Math. Soc., Providence, RI, (2004), 103-134.

[6]

A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Diff. Equations, 30 (2005), 67-96.

show all references

References:
[1]

I. Beltiţă and A. Melin, Analysis of the quadratic term in the backscattering transformation, Math. Scand., 105 (2009), 218-234.

[2]

I. Beltiţă and A. Melin, Local smoothing for the backscattering transformation, Comm. Partial Diff. Equations, 34 (2009), 233-256.

[3]

L. Hörmander, "The Analysis of Linear Partial Differential Operators'' (I-IV), Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1983-1985.

[4]

A. Melin, Smoothness of higher order terms in backscattering, in "Wave Phenomena and Asymptotic Analysis,'' RIMS Kokyuroku, 1315 (2003), 43-51.

[5]

A. Melin, Some transforms in potential scattering in odd dimension, in "Inverse Problems and Spectral Theory,'' Contemp. Math., 348, Amer. Math. Soc., Providence, RI, (2004), 103-134.

[6]

A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Diff. Equations, 30 (2005), 67-96.

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