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

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  • 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.
    Mathematics Subject Classification: Primary: 81U025; Secondary: 35R30.


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


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


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


    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.


    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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