Inverse scattering without phase information

R.G. Novikov

doi:10.5802/slsedp.74

R.G. Novikov

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

Abstract

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.

Published online: 2015-12-15

DOI: 10.5802/slsedp.74

@article{SLSEDP_2014-2015____A16_0,
     author = {R.G. Novikov},
     title = {Inverse scattering without phase information},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:16},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2014-2015},
     doi = {10.5802/slsedp.74},
     language = {en},
     url = {https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.74/}
}

TY  - JOUR
TI  - Inverse scattering without phase information
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:16
PY  - 2014-2015
DA  - 2014-2015///
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.74/
UR  - https://doi.org/10.5802/slsedp.74
DO  - 10.5802/slsedp.74
LA  - en
ID  - SLSEDP_2014-2015____A16_0
ER  -

%0 Journal Article
%T Inverse scattering without phase information
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:16
%D 2014-2015
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://doi.org/10.5802/slsedp.74
%R 10.5802/slsedp.74
%G en
%F SLSEDP_2014-2015____A16_0

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

References
Cited by

[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.

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

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

[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.

[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).

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

[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.

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

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

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

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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. doi:10.1515/jiip-2015-0025; arXiv:1412.8210v1, December 28, 2014.

[KR2] M.V. Klibanov, V.G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, arXiv:1505.01905v1, May 8, 2015.

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

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

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

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

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

[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.

[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.

[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.

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

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

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

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

[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.

[N9] R.G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geom. Anal. doi:10.1007/s12220-014-9553-7; arXiv:1412.5006v1, December 16, 2014.

[N10] R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math. doi:10.1016/j.bulsci.2015.04.005; arXiv:1502.02282v2, February 14, 2015.

[N11] R.G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian Journal of Mathematical and Computer Applications 3(1), 2015, 63-69; arXiv:1503.02159, March 7, 2015.

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

[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.

[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.

Cited by Sources: