A New Convergent Algorithm to Approximate Potentials from Fixed Angle Scattering Data

We introduce a new iterative method to recover a real compact supported potential of the Schodinger operator from their fixed angle scattering data. The method combines a fixed point argument with ...

[1]  S. Moskow,et al.  Convergence of the Born and inverse Born series for electromagnetic scattering , 2017 .

[2]  S. Moskow,et al.  Convergence and stability of the inverse scattering series for diffuse waves , 2008, 0804.2681.

[3]  Reconstruction of Singularities of a Scattering Potential in Two Dimensions , 1994 .

[4]  A. Komech.  Introduction to the Scattering Theory for the Schrödinger Equation ( The Agmon-Jensen-Kato approach ) , 2022 .

[5]  G. Mockenhaupt.  A restriction theorem for the Fourier transform , 1991 .

[6]  Reese T. Prosser,et al.  Formal solutions of inverse scattering problems. III. , 1969 .

[7]  G. Eskin,et al.  Inverse backscattering in two dimensions , 1991 .

[8]  V. Serov,et al.  Recovery of Singularities of a Multidimensional Scattering Potential , 1998 .

[9]  G. Eskin.  Lectures on Linear Partial Differential Equations , 2011 .

[10]  Shmuel Agmon,et al.  Spectral properties of Schrödinger operators and scattering theory , 1975 .

[11]  John C. Schotland,et al.  INVERSE BORN SERIES FOR SCALAR WAVES , 2012 .

[12]  G. Eskin,et al.  Inverse backscattering , 1992 .

[13]  J. Barceló,et al.  Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers , 2010 .

[14]  Christopher D. Sogge,et al.  Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators , 1987 .

[15]  G. Eskin,et al.  The inverse backscattering problem in three dimensions , 1989 .

[16]  A. Ruiz.  RECOVERY OF THE SINGULARITIES OF A POTENTIAL FROM FIXED ANGLE SCATTERING DATA , 2001 .

[17]  John C. Schotland,et al.  Numerical studies of the inverse Born series for diffuse waves , 2009 .

[18]  J. L. Zolesio,et al.  Multiplication dans les espaces de Besov , 1977, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  M. Murata.  Asymptotic expansions in time for solutions of Schrödinger-type equations , 1982 .

[20]  K. Mochizuki.  Eigenfunction Expansions Associated with the Schrödinger Operator with a Complex Potential and the Scattering Theory , 1968 .

[21]  Reese T. Prosser,et al.  Formal solutions of inverse scattering problems. IV. Error estimates , 1982 .

[22]  A. Komech,et al.  Dispersion Decay and Scattering Theory , 2012 .

[23]  I. Gel'fand,et al.  On the determination of a differential equation from its spectral function , 1955 .

[24]  Generic Uniqueness for Two Inverse Problems in POtential Scattering , 1992 .

[25]  H. Moses,et al.  Calculation of the Scattering Potential from Reflection Coefficients , 1956 .

[26]  C. Castro,et al.  Numerical approximation of the potential in the two-dimesional inverse scattering problem , 2015, 1507.07827.

[27]  Walter Kohn,et al.  Construction of a Potential from a Phase Shift , 1952 .

[28]  L. Vega,et al.  Weighted Estimates for the Helmholtz Equation and Some Applications , 1997 .

[29]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[30]  L. Vega,et al.  On local regularity of Schrödinger equations , 1993 .

[31]  Arne Jensen,et al.  Spectral properties of Schrödinger operators and time-decay of the wave functions , 1979 .

[32]  J. González.  Problema inverso de scattering para la ecuación de Schrödinger: reconstrucción parcial del potencial a partir de datos de retrodispersión en 2D y 3D , 2007 .

[33]  Inversion of discontinuities for the Schro¨dinger equation in three dimensions , 1991 .

[34]  Gunther Uhlmann,et al.  Recovering singularities of a potential from singularities of scattering data , 1993 .

[35]  Elias M. Stein,et al.  Unique continuation and absence of positive eigenvalues for Schrodinger operators , 1985 .

[36]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .