# American Institute of Mathematical Sciences

May  2012, 6(2): 321-355. doi: 10.3934/ipi.2012.6.321

## Reconstruction of the singularities of a potential from backscattering data in 2D and 3D

 1 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland 2 Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049-Madrid

Received  June 2011 Revised  December 2011 Published  May 2012

We prove that the singularities of a potential in two and three dimensional Schrödinger equation are the same as those of the Born approximation (Diffraction Tomography), obtained from backscattering inverse data, with an accuracy of $1/2^-$ derivative in the scale of $L^2$-based Sobolev spaces. This improves previous results, see [30] and [20], removing several constrains on the a priori regularity of the potential. The improvement is based on the study of the smoothing properties of the quartic term in the Neumann-Born expansion of the scattering amplitude in 3D, together with a Leibniz formula for multiple scattering valid in any dimension.
Citation: Juan Manuel Reyes, Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D. Inverse Problems & Imaging, 2012, 6 (2) : 321-355. doi: 10.3934/ipi.2012.6.321
##### References:
 [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa (4), II (1975), 151. Google Scholar [2] J. A. Barceló, D. Faraco, A. Ruiz and A. Vargas, Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers,, Math. Ann., 346 (2010), 505. doi: 10.1007/s00208-009-0398-5. Google Scholar [3] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a casual generalized Radon transform,, J. Math. Phys., 26 (1985), 99. doi: 10.1063/1.526755. Google Scholar [4] I. Beltita and A. Mellin, Analysis of the quadratic term in the backscattering transform,, Math. Scand., 105 (2009), 218. Google Scholar [5] I. Beltita and A. Mellin, Local smoothing for the backscattering transform,, Comm. Partial Differential Equations, 34 (2009), 233. doi: 10.1080/03605300902812384. Google Scholar [6] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'', John Wiley & Sons, (1983). Google Scholar [7] G. Eskin and J. Ralston, The inverse backscattering problem in 3 dimension,, Comm. Math. Phys., 124 (1989), 169. doi: 10.1007/BF01219194. Google Scholar [8] G. Eskin and J. Ralston, Inverse backscattering in two dimensions,, Comm. Math. Phys., 138 (1991), 451. doi: 10.1007/BF02102037. Google Scholar [9] G. Eskin and J. Ralston, Inverse backscattering,, J. Anal. Math., 58 (1992), 177. doi: 10.1007/BF02790363. Google Scholar [10] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'', Pitman Boston, (1985). Google Scholar [11] A. Greenleaf and G. Uhlmann, Recovery of singularities of a potential from singularities of the scattering data,, Comm. Math. Phys., 157 (1993), 549. doi: 10.1007/BF02096882. Google Scholar [12] P. Hajlasz, Sobolev spaces on an arbitrary metric space,, Potential Anal., 5 (1996), 403. Google Scholar [13] C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficients differential operators,, Duke Math. J., 55 (1987), 329. doi: 10.1215/S0012-7094-87-05518-9. Google Scholar [14] R. Lagergren, "Backscattering in Three Dimensions,'', Ph.D thesis, (2001). Google Scholar [15] R. Lagergren, The back-scattering problem in three dimensions,, J. Pseudo-Differ. Oper. Appl., 2 (2011), 1. doi: 10.1007/s11868-010-0021-2. Google Scholar [16] A. Melin, Some transforms in potential scattering in odd dimension,, in, 348 (2004), 103. Google Scholar [17] R. Melrose and G. Uhlmann, Generalized backscattering and the Lax-Phillips transform,, Serdica Math. J., 34 (2008), 355. Google Scholar [18] A. Nachman, Inverse scattering at fixed energy,, in, (1992), 434. Google Scholar [19] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta\Psi+(v(x)-Eu(x))\Psi=0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar [20] P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions,, Comm. Partial Differential Equations, 26 (2001), 697. doi: 10.1081/PDE-100001768. Google Scholar [21] L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential,, SIAM J. Math. Anal., 29 (1998), 697. doi: 10.1137/S0036141096305796. Google Scholar [22] L. Päivärinta, V. Serov and E. Somersalo, Reconstruction of singularities of a scattering potential in two dimensions,, Adv. in Appl. Math., 15 (1994), 97. doi: 10.1006/aama.1994.1003. Google Scholar [23] L. Päivärinta and E. Somersalo, Inversion of discontinuities for the Schrödinger equation in three dimensions,, SIAM J. Math. Anal., 22 (1991), 480. doi: 10.1137/0522031. Google Scholar [24] R. T. Prosser, Formal solutions of inverse scattering problems,, J. Math. Phys., 23 (1982), 2127. doi: 10.1063/1.525267. Google Scholar [25] A. G. Ramm, Recovery of a potential from fixed-energy scattering data,, Inverse Problems, 4 (1988), 877. Google Scholar [26] J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D,, Inverse Problems, 23 (2007), 625. doi: 10.1088/0266-5611/23/2/010. Google Scholar [27] J. M. Reyes, "Problema Inverso de Scattering para la Ecuación de Schrödinger: Reconstrucción Parcial del Potencial a Partir de Datos de Retrodispersión en 2D y 3D,'', (Spanish), (2007). Google Scholar [28] A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data,, Comm. Partial Differential Equations, 26 (2001), 1721. Google Scholar [29] A. Ruiz, "Harmonic Analysis and Inverse Problems,'', Notes of the 4th Summer School in Inverse Problems, (2002). Google Scholar [30] A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Differential Equations, 30 (2005), 67. doi: 10.1081/PDE-200044450. Google Scholar [31] P. Stefanov, Generic uniqueness for two inverse problems in potential scattering,, Comm. Partial Differential Equations, 17 (1992), 55. doi: 10.1080/03605309208820834. Google Scholar [32] Z. Sun and G. Uhlmann, Generic uniqueness for an inverse boundary value problem,, Duke Math. J., 62 (1991), 131. doi: 10.1215/S0012-7094-91-06206-X. Google Scholar [33] G. Uhlmann, A time-dependent approach to the inverse backscattering problem,, Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 17 (2001), 703. Google Scholar [34] G. N. Watson, "The Theory of Bessel Functions,'', Cambridge University Press, (1948). Google Scholar

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##### References:
 [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa (4), II (1975), 151. Google Scholar [2] J. A. Barceló, D. Faraco, A. Ruiz and A. Vargas, Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers,, Math. Ann., 346 (2010), 505. doi: 10.1007/s00208-009-0398-5. Google Scholar [3] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a casual generalized Radon transform,, J. Math. Phys., 26 (1985), 99. doi: 10.1063/1.526755. Google Scholar [4] I. Beltita and A. Mellin, Analysis of the quadratic term in the backscattering transform,, Math. Scand., 105 (2009), 218. Google Scholar [5] I. Beltita and A. Mellin, Local smoothing for the backscattering transform,, Comm. Partial Differential Equations, 34 (2009), 233. doi: 10.1080/03605300902812384. Google Scholar [6] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'', John Wiley & Sons, (1983). Google Scholar [7] G. Eskin and J. Ralston, The inverse backscattering problem in 3 dimension,, Comm. Math. Phys., 124 (1989), 169. doi: 10.1007/BF01219194. Google Scholar [8] G. Eskin and J. Ralston, Inverse backscattering in two dimensions,, Comm. Math. Phys., 138 (1991), 451. doi: 10.1007/BF02102037. Google Scholar [9] G. Eskin and J. Ralston, Inverse backscattering,, J. Anal. Math., 58 (1992), 177. doi: 10.1007/BF02790363. Google Scholar [10] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'', Pitman Boston, (1985). Google Scholar [11] A. Greenleaf and G. Uhlmann, Recovery of singularities of a potential from singularities of the scattering data,, Comm. Math. Phys., 157 (1993), 549. doi: 10.1007/BF02096882. Google Scholar [12] P. Hajlasz, Sobolev spaces on an arbitrary metric space,, Potential Anal., 5 (1996), 403. Google Scholar [13] C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficients differential operators,, Duke Math. J., 55 (1987), 329. doi: 10.1215/S0012-7094-87-05518-9. Google Scholar [14] R. Lagergren, "Backscattering in Three Dimensions,'', Ph.D thesis, (2001). Google Scholar [15] R. Lagergren, The back-scattering problem in three dimensions,, J. Pseudo-Differ. Oper. Appl., 2 (2011), 1. doi: 10.1007/s11868-010-0021-2. Google Scholar [16] A. Melin, Some transforms in potential scattering in odd dimension,, in, 348 (2004), 103. Google Scholar [17] R. Melrose and G. Uhlmann, Generalized backscattering and the Lax-Phillips transform,, Serdica Math. J., 34 (2008), 355. Google Scholar [18] A. Nachman, Inverse scattering at fixed energy,, in, (1992), 434. Google Scholar [19] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta\Psi+(v(x)-Eu(x))\Psi=0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar [20] P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions,, Comm. Partial Differential Equations, 26 (2001), 697. doi: 10.1081/PDE-100001768. Google Scholar [21] L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential,, SIAM J. Math. Anal., 29 (1998), 697. doi: 10.1137/S0036141096305796. Google Scholar [22] L. Päivärinta, V. Serov and E. Somersalo, Reconstruction of singularities of a scattering potential in two dimensions,, Adv. in Appl. Math., 15 (1994), 97. doi: 10.1006/aama.1994.1003. Google Scholar [23] L. Päivärinta and E. Somersalo, Inversion of discontinuities for the Schrödinger equation in three dimensions,, SIAM J. Math. Anal., 22 (1991), 480. doi: 10.1137/0522031. Google Scholar [24] R. T. Prosser, Formal solutions of inverse scattering problems,, J. Math. Phys., 23 (1982), 2127. doi: 10.1063/1.525267. Google Scholar [25] A. G. Ramm, Recovery of a potential from fixed-energy scattering data,, Inverse Problems, 4 (1988), 877. Google Scholar [26] J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D,, Inverse Problems, 23 (2007), 625. doi: 10.1088/0266-5611/23/2/010. Google Scholar [27] J. M. Reyes, "Problema Inverso de Scattering para la Ecuación de Schrödinger: Reconstrucción Parcial del Potencial a Partir de Datos de Retrodispersión en 2D y 3D,'', (Spanish), (2007). Google Scholar [28] A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data,, Comm. Partial Differential Equations, 26 (2001), 1721. Google Scholar [29] A. Ruiz, "Harmonic Analysis and Inverse Problems,'', Notes of the 4th Summer School in Inverse Problems, (2002). Google Scholar [30] A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Differential Equations, 30 (2005), 67. doi: 10.1081/PDE-200044450. Google Scholar [31] P. Stefanov, Generic uniqueness for two inverse problems in potential scattering,, Comm. Partial Differential Equations, 17 (1992), 55. doi: 10.1080/03605309208820834. Google Scholar [32] Z. Sun and G. Uhlmann, Generic uniqueness for an inverse boundary value problem,, Duke Math. J., 62 (1991), 131. doi: 10.1215/S0012-7094-91-06206-X. Google Scholar [33] G. Uhlmann, A time-dependent approach to the inverse backscattering problem,, Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 17 (2001), 703. Google Scholar [34] G. N. Watson, "The Theory of Bessel Functions,'', Cambridge University Press, (1948). Google Scholar
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