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

Previous Article
Inverse problems for quantum trees II: Recovering matching conditions for star graphs
IPI Home
This Issue
Next Article
Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform

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

Abstract
Related Papers

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 & 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. Google Scholar
[2]	I. Beltiţă and A. Melin, Local smoothing for the backscattering transformation,, Comm. Partial Diff. Equations, 34 (2009), 233. Google Scholar
[3]	L. Hörmander, "The Analysis of Linear Partial Differential Operators'' (I-IV),, Springer Verlag, (): 1983. Google Scholar
[4]	A. Melin, Smoothness of higher order terms in backscattering,, in, 1315 (2003), 43. Google Scholar
[5]	A. Melin, Some transforms in potential scattering in odd dimension,, in, 348 (2004), 103. Google Scholar
[6]	A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Diff. Equations, 30 (2005), 67. Google Scholar

show all references

References:

[1]	I. Beltiţă and A. Melin, Analysis of the quadratic term in the backscattering transformation,, Math. Scand., 105 (2009), 218. Google Scholar
[2]	I. Beltiţă and A. Melin, Local smoothing for the backscattering transformation,, Comm. Partial Diff. Equations, 34 (2009), 233. Google Scholar
[3]	L. Hörmander, "The Analysis of Linear Partial Differential Operators'' (I-IV),, Springer Verlag, (): 1983. Google Scholar
[4]	A. Melin, Smoothness of higher order terms in backscattering,, in, 1315 (2003), 43. Google Scholar
[5]	A. Melin, Some transforms in potential scattering in odd dimension,, in, 348 (2004), 103. Google Scholar
[6]	A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Diff. Equations, 30 (2005), 67. Google Scholar

[1]	Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844
[2]	Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291
[3]	Enrico Capobianco. Born to be big: Data, graphs, and their entangled complexity. Big Data & Information Analytics, 2016, 1 (2&3) : 163-169. doi: 10.3934/bdia.2016002
[4]	Lyndsey Clark. The $\beta$-transformation with a hole. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1249-1269. doi: 10.3934/dcds.2016.36.1249
[5]	Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[6]	Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353
[7]	Bin Cheng, Alex Mahalov. Time-averages of fast oscillatory systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1151-1162. doi: 10.3934/dcdss.2013.6.1151
[8]	Alexander Barg, Oleg R. Musin. Codes in spherical caps. Advances in Mathematics of Communications, 2007, 1 (1) : 131-149. doi: 10.3934/amc.2007.1.131
[9]	Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905
[10]	Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010
[11]	Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems & Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010
[12]	Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803
[13]	Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-31. doi: 10.3934/dcdsb.2019063
[14]	Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359
[15]	Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310
[16]	Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119
[17]	Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100
[18]	Shaobo Lin, Xingping Sun, Zongben Xu. Discretizing spherical integrals and its applications. Conference Publications, 2013, 2013 (special) : 499-514. doi: 10.3934/proc.2013.2013.499
[19]	Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243
[20]	Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences