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 and 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), 151-218.

[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), 505-544. doi: 10.1007/s00208-009-0398-5.

[3]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a casual generalized Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.

[4]

I. Beltita and A. Mellin, Analysis of the quadratic term in the backscattering transform, Math. Scand., 105 (2009), 218-234.

[5]

I. Beltita and A. Mellin, Local smoothing for the backscattering transform, Comm. Partial Differential Equations, 34 (2009), 233-256. doi: 10.1080/03605300902812384.

[6]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'' John Wiley & Sons, New York, 1983.

[7]

G. Eskin and J. Ralston, The inverse backscattering problem in 3 dimension, Comm. Math. Phys., 124 (1989), 169-215. doi: 10.1007/BF01219194.

[8]

G. Eskin and J. Ralston, Inverse backscattering in two dimensions, Comm. Math. Phys., 138 (1991), 451-486. doi: 10.1007/BF02102037.

[9]

G. Eskin and J. Ralston, Inverse backscattering, J. Anal. Math., 58 (1992), 177-190. doi: 10.1007/BF02790363.

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Pitman Boston, 1985.

[11]

A. Greenleaf and G. Uhlmann, Recovery of singularities of a potential from singularities of the scattering data, Comm. Math. Phys., 157 (1993), 549-572. doi: 10.1007/BF02096882.

[12]

P. Hajlasz, Sobolev spaces on an arbitrary metric space, Potential Anal., 5 (1996), 403-415.

[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), 329-347. doi: 10.1215/S0012-7094-87-05518-9.

[14]

R. Lagergren, "Backscattering in Three Dimensions,'' Ph.D thesis, Lund University, 2001.

[15]

R. Lagergren, The back-scattering problem in three dimensions, J. Pseudo-Differ. Oper. Appl., 2 (2011), no. 1, 1-64. doi: 10.1007/s11868-010-0021-2.

[16]

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.

[17]

R. Melrose and G. Uhlmann, Generalized backscattering and the Lax-Phillips transform, Serdica Math. J., 34 (2008), 355-372.

[18]

A. Nachman, Inverse scattering at fixed energy, in "Proceedings of the 10th International Congress on Mathematical Physics," Leipzig, Springer Verlag, (1992), 434-441.

[19]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta\Psi+(v(x)-Eu(x))\Psi=0$, Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.

[20]

P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. Partial Differential Equations, 26 (2001), 697-715. doi: 10.1081/PDE-100001768.

[21]

L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711. doi: 10.1137/S0036141096305796.

[22]

L. Päivärinta, V. Serov and E. Somersalo, Reconstruction of singularities of a scattering potential in two dimensions, Adv. in Appl. Math., 15 (1994), 97-113. doi: 10.1006/aama.1994.1003.

[23]

L. Päivärinta and E. Somersalo, Inversion of discontinuities for the Schrödinger equation in three dimensions, SIAM J. Math. Anal., 22 (1991), 480-499. doi: 10.1137/0522031.

[24]

R. T. Prosser, Formal solutions of inverse scattering problems, J. Math. Phys., 23 (1982), 2127-2130. doi: 10.1063/1.525267.

[25]

A. G. Ramm, Recovery of a potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886.

[26]

J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D, Inverse Problems, 23 (2007), 625-643. doi: 10.1088/0266-5611/23/2/010.

[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), Ph.D thesis, Universidad Autónoma de Madrid, 2007. Available from: http://www.uam.es/gruposinv/inversos/publicaciones/index.html.

[28]

A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data, Comm. Partial Differential Equations, 26 (2001), 1721-1738.

[29]

A. Ruiz, "Harmonic Analysis and Inverse Problems,'' Notes of the 4th Summer School in Inverse Problems, Oulu, Finland, 2002. Available from: http://www.uam.es/gruposinv/inversos/publicaciones/index.html.

[30]

A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Differential Equations, 30 (2005), 67-96. doi: 10.1081/PDE-200044450.

[31]

P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations, 17 (1992), 55-68. doi: 10.1080/03605309208820834.

[32]

Z. Sun and G. Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J., 62 (1991), 131-155. doi: 10.1215/S0012-7094-91-06206-X.

[33]

G. Uhlmann, A time-dependent approach to the inverse backscattering problem, Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000), Inverse Problems, 17 (2001), 703-716.

[34]

G. N. Watson, "The Theory of Bessel Functions,'' Cambridge University Press, New York, 1948.

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa (4), II (1975), 151-218.

[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), 505-544. doi: 10.1007/s00208-009-0398-5.

[3]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a casual generalized Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.

[4]

I. Beltita and A. Mellin, Analysis of the quadratic term in the backscattering transform, Math. Scand., 105 (2009), 218-234.

[5]

I. Beltita and A. Mellin, Local smoothing for the backscattering transform, Comm. Partial Differential Equations, 34 (2009), 233-256. doi: 10.1080/03605300902812384.

[6]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,'' John Wiley & Sons, New York, 1983.

[7]

G. Eskin and J. Ralston, The inverse backscattering problem in 3 dimension, Comm. Math. Phys., 124 (1989), 169-215. doi: 10.1007/BF01219194.

[8]

G. Eskin and J. Ralston, Inverse backscattering in two dimensions, Comm. Math. Phys., 138 (1991), 451-486. doi: 10.1007/BF02102037.

[9]

G. Eskin and J. Ralston, Inverse backscattering, J. Anal. Math., 58 (1992), 177-190. doi: 10.1007/BF02790363.

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Pitman Boston, 1985.

[11]

A. Greenleaf and G. Uhlmann, Recovery of singularities of a potential from singularities of the scattering data, Comm. Math. Phys., 157 (1993), 549-572. doi: 10.1007/BF02096882.

[12]

P. Hajlasz, Sobolev spaces on an arbitrary metric space, Potential Anal., 5 (1996), 403-415.

[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), 329-347. doi: 10.1215/S0012-7094-87-05518-9.

[14]

R. Lagergren, "Backscattering in Three Dimensions,'' Ph.D thesis, Lund University, 2001.

[15]

R. Lagergren, The back-scattering problem in three dimensions, J. Pseudo-Differ. Oper. Appl., 2 (2011), no. 1, 1-64. doi: 10.1007/s11868-010-0021-2.

[16]

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.

[17]

R. Melrose and G. Uhlmann, Generalized backscattering and the Lax-Phillips transform, Serdica Math. J., 34 (2008), 355-372.

[18]

A. Nachman, Inverse scattering at fixed energy, in "Proceedings of the 10th International Congress on Mathematical Physics," Leipzig, Springer Verlag, (1992), 434-441.

[19]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta\Psi+(v(x)-Eu(x))\Psi=0$, Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.

[20]

P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. Partial Differential Equations, 26 (2001), 697-715. doi: 10.1081/PDE-100001768.

[21]

L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711. doi: 10.1137/S0036141096305796.

[22]

L. Päivärinta, V. Serov and E. Somersalo, Reconstruction of singularities of a scattering potential in two dimensions, Adv. in Appl. Math., 15 (1994), 97-113. doi: 10.1006/aama.1994.1003.

[23]

L. Päivärinta and E. Somersalo, Inversion of discontinuities for the Schrödinger equation in three dimensions, SIAM J. Math. Anal., 22 (1991), 480-499. doi: 10.1137/0522031.

[24]

R. T. Prosser, Formal solutions of inverse scattering problems, J. Math. Phys., 23 (1982), 2127-2130. doi: 10.1063/1.525267.

[25]

A. G. Ramm, Recovery of a potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886.

[26]

J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D, Inverse Problems, 23 (2007), 625-643. doi: 10.1088/0266-5611/23/2/010.

[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), Ph.D thesis, Universidad Autónoma de Madrid, 2007. Available from: http://www.uam.es/gruposinv/inversos/publicaciones/index.html.

[28]

A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data, Comm. Partial Differential Equations, 26 (2001), 1721-1738.

[29]

A. Ruiz, "Harmonic Analysis and Inverse Problems,'' Notes of the 4th Summer School in Inverse Problems, Oulu, Finland, 2002. Available from: http://www.uam.es/gruposinv/inversos/publicaciones/index.html.

[30]

A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Differential Equations, 30 (2005), 67-96. doi: 10.1081/PDE-200044450.

[31]

P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations, 17 (1992), 55-68. doi: 10.1080/03605309208820834.

[32]

Z. Sun and G. Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J., 62 (1991), 131-155. doi: 10.1215/S0012-7094-91-06206-X.

[33]

G. Uhlmann, A time-dependent approach to the inverse backscattering problem, Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000), Inverse Problems, 17 (2001), 703-716.

[34]

G. N. Watson, "The Theory of Bessel Functions,'' Cambridge University Press, New York, 1948.

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