ИСТИНА

We consider direct and inverse scattering for the Schroedinger equation of quantum mechanics and for the Helmholtz equation of acoustics or electrodynamics. In addition, only scattering data without phase information can be measured directly in practice in quantum mechanics and in some other cases. Note that in quantum mechanics this limitation is related to the probabilistic interpretation of the wavefunction proposed originally by Max Born in 1926. In this connection we report on non-uniqueness, uniqueness and reconstruction results for inverse scattering without phase information. We are motivated by recent and very essential progress in this domain. In particular, in the first part of this talk we present the results of [N1], [N2], [N3] and in the second part of the talk we present the results of [AN] and [AGHN]. For more information, see [AN], [AGHN], [N1], [N2], [N3], and references therein. In particular, among preceeding works we would like to mention [AS], [GPS], [K]. References [AN] A. D. Agaltsov, R. G. Novikov, Error estimates for phaseless inverse scattering in the Born approximation at high energies, The Journal of Geometric Analysis, to appear; arXiv:1604.06555v2 [AGHN] A. D. Agaltsov, A. Gillman, T. Hohage, R. G. Novikov, An iterative approach to monochromatic phaseless inverse scaterring, in preparation [AS] T. Aktosun, P. Sacks, Inverse problem on the line without phase information, Inverse Problems 14, 1998, 211-224 [GPS] A. A. Govyadinov, G. Y. Panasyuk, J. C. Schotland, Phaseless three-dimensional optical nanoimaging, Physical Review Letters 103, 2009, 213901 [K] M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math. 74 (2), 2014, 392-410 [N1] R. G. Novikov, Phaseless inverse scattering in the one dimensional case, Eurasian Journal of Mathematical and Computer Applications 3 (1), 2015, 64-70 [N2] R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin Des Sciences Mathematiques 139 (8), 2015, 23-936 [N3] R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, The Journal of Geometric Analysis 26 (1), 2016, 346-359