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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

 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.
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.

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##### 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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