Inverse scattering without phase information

R.G. Novikov

doi:10.5802/slsedp.74

¹CNRS (UMR 7641), Centre de Mathématiques Appliquées,École Polytechnique, 91128 Palaiseau, France & IEPT RAS, 117997 Moscow, Russia

Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 16, 13 p.

Résumé

We report on non-uniqueness, uniqueness and reconstruction results in quantum mechanical and acoustic inverse scattering without phase information. We are motivated by recent and very essential progress in this domain.

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DOI : 10.5802/slsedp.74

Affiliations des auteurs :

R.G. Novikov ¹

¹ CNRS (UMR 7641), Centre de Mathématiques Appliquées,École Polytechnique, 91128 Palaiseau, France & IEPT RAS, 117997 Moscow, Russia

R.G. Novikov. Inverse scattering without phase information. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 16, 13 p.. doi: 10.5802/slsedp.74

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Bibliographie
Cité par

[AS] T. Aktosun, P.E. Sacks, Inverse problem on the line without phase information, Inverse Problems 14, 1998, 211-224. | Zbl | MR

[AW] T. Aktosun, R. Weder, Inverse scattering with partial information on the potential, J. Math. Anal. Appl. 270, 2002, 247-266. | Zbl | MR

[ABR] N.V. Alexeenko, V.A. Burov, O.D. Rumyantseva, Solution of the three-dimensional acoustical inverse scattering problem. The modified Novikov algorithm, Acoust. J. 54(3), 2008, 469-482 (in Russian); English transl.: Acoust. Phys. 54(3), 2008, 407-419.

[Ber] Yu.M. Berezanskii, On the uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation, Tr. Mosk. Mat. Obshch. 7, 1958, 3-62 (in Russian). | Zbl | MR

[BS] F.A. Berezin, M.A. Shubin, The Schrödinger Equation, Vol. 66 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1991. | Zbl | MR

[Buc] A.L. Buckhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16(1), 2008, 19-33. | Zbl | MR

[BAR] V.A. Burov, N.V. Alekseenko, O.D. Rumyantseva, Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem, Acoust. J. 55(6), 2009, 784-798 (in Russian); English transl.: Acoustical Physics 55(6), 2009, 843-856.

[ChS] K. Chadan, P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer, Berlin, 1989. | Zbl | MR

[EW] V.Enss, R.Weder, Inverse potential scattering: a geometrical approach,

[DT] P. Deift, E.Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32, 1979, 121-251. | Zbl | MR

[E] G. Eskin, Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics, Vol.123, American Mathematical Society, 2011. | Zbl | MR

[F1] L.D. Faddeev, Uniqueness of the solution of the inverse scattering problem, Vest. Leningrad Univ. 7, 1956, 126-130 (in Russian). | MR

[F2] L.D. Faddeev, Inverse problem of quantum scattering theory. II, Itogy Nauki i Tekh. Ser. Sovrem. Probl. Mat. 3, 1974, 93-180 (in Russian); English transl.: Journal of Soviet Mathematics 5, 1976, 334-396. | Zbl | MR

[FM] L.D. Faddeev, S.P. Merkuriev, Quantum Scattering Theory for Multi-particle Systems, Nauka, Moscow, 1985 (in Russian); English transl: Math. Phys. Appl. Math. 11 (1993), Kluwer Academic Publishers Group, Dordrecht. | MR

[GS] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential. I. The case of an a.c. component in the spectrum, Helv. Phys. Acta 70, 1997, 66-71. | Zbl | MR

[G] P.G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, Uspekhi Mat. Nauk 55:6(336),2000, 3-70 (Russian); English transl.: Russian Math. Surveys 55:6, 2000, 1015-1083. | Zbl | MR

[HH] P. Hähner, T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problem and applications, SIAM J. Math. Anal., 33(3), 2001, 670-685. | Zbl | MR

[HN] G.M. Henkin, R.G. Novikov, The $\bar{\partial}$ -equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 42(3), 1987, 93-152 (in Russian); English transl.: Russ. Math. Surv. 42(3), 1987, 109-180. | Zbl | MR

[I] M.I. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Funkt. Anal. Prilozhen. 47(3), 2013, 28-36 (in Russian); English transl.: Funct. Anal Appl. 47, 2013, 187-194. | MR

[IN] M.I. Isaev, R.G. Novikov, New global stability estimates for monochromatic inverse acoustic scattering, SIAM J. Math. Anal. 45(3), 2013, 1495-1504. | Zbl | MR

[K] M.V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math. 74, 2014, 392-410. | Zbl | MR

[KR1] M.V. Klibanov, V.G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse Ill-Posed Probl. ; , December 28, 2014. | DOI | arXiv | MR

[KR2] M.V. Klibanov, V.G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, , May 8, 2015. | arXiv | MR

[KS] M.V. Klibanov, P.E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Phys. 33, 1992, 3813-3821. | Zbl | MR

[L] B.M. Levitan, Inverse Sturm-Liuville Problems, VSP, Zeist, 1987. | Zbl | MR

[Mar] V.A. Marchenko, Sturm-Liuville Operators and Applications, Birkhäuser, Basel, 1986. | Zbl | MR

[Mel] R.B. Melrose, Geometric scattering theory, Cambridge University Press, 1995. | Zbl | MR

[Mos] H.E. Moses, Calculation of the scattering potential from reflection coefficients, Phys. Rev. 102, 1956, 559-567. | Zbl | MR

[New] R.G. Newton, Inverse Schrödinger scattering in three dimensions, Springer, Berlin, 1989. | Zbl | MR

[N1] R.G. Novikov, Multidimensional inverse spectral problem for the equation $- Δ ψ + (v (x) - E u (x)) ψ = 0$ , Funkt. Anal. Prilozhen. 22(4), 1988, 11-22 (in Russian); English transl.: Funct. Anal. Appl. 22, 1988, 263-272. | Zbl | MR

[N2] R.G. Novikov, The inverse scattering problem at fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal., 103, 1992, 409-463. | Zbl | MR

[N3] R.G. Novikov, The inverse scattering problem at fixed energy for Schrödinger equation with an exponentially decreasing potential, Comm. Math. Phys. 161, 1994, 569-595. | Zbl | MR

[N4] R.G. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension 1, Bull. Sci. Math. 120, 1996, 473-491. | Zbl | MR

[N5] R.G. Novikov, Approximate inverse quantum scattering at fixed energy in dimension 2, Proc. Steklov Inst. Math. 225, 1999, 285-302. | Zbl | MR

[N6] R.G. Novikov, The $\bar{\partial}$ -approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal. 18, 2008, 612-631. | Zbl | MR

[N7] R.G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy, Physics Letters A 375, 2011, 1233-1235. | Zbl | MR

[N8] R.G. Novikov, An iterative approach to non-overdetermined inverse scattering at fixed energy, Mat. Sb. 206(1), 2015, 131-146 (in Russian); English transl.: Sbornik: Mathematics 206(1), 2015, 120-134. | MR

[N9] R.G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geom. Anal. ; , December 16, 2014. | DOI | arXiv

[N10] R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math. ; , February 14, 2015. | DOI | arXiv | MR

[N11] R.G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian Journal of Mathematical and Computer Applications 3(1), 2015, 63-69; , March 7, 2015. | arXiv | MR

[NM] N.N. Novikova, V.M. Markushevich, On the uniqueness of the solution of the inverse scattering problem on the real axis for the potentials placed on the positive half-axis. Comput. Seismology 18, 1985, 176-184 (in Russian).

[R] T. Regge, Introduction to complex orbital moments, Nuovo Cimento 14, 1959, 951-976. | Zbl | MR

[S] P. Stefanov, Stability of the inverse problem in potential scattering at fixed energy, Annales de l’Institut Fourier, tome 40(4), 1990, 867-884. | Zbl | MR | Numdam

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