TY - GEN
T1 - Multidimensional Inverse Scattering for the Schrödinger Equation
AU - Novikov, Roman G.
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation. A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice. In this connection our considerations include reconstructions from non-overdetermined monochromatic scattering data and formulas for phase recovering from phaseless scattering data. Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles. This paper is based, in particular, on results going back to M. Born (1926), L. Faddeev (1956, 1974), S. Manakov (1981), R. Beals, R. Coifman (1985), P. Grinevich, R. Novikov (1986), G. Henkin, R. Novikov (1987), and on more recent results of R. Novikov (1998–2019), A. Agaltsov, T. Hohage, R. Novikov (2019). This paper is an extended version of the talk given at the 12th ISAAC Congress, Aveiro, Portugal, 29 July–2 August, 2019.
AB - We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation. A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice. In this connection our considerations include reconstructions from non-overdetermined monochromatic scattering data and formulas for phase recovering from phaseless scattering data. Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles. This paper is based, in particular, on results going back to M. Born (1926), L. Faddeev (1956, 1974), S. Manakov (1981), R. Beals, R. Coifman (1985), P. Grinevich, R. Novikov (1986), G. Henkin, R. Novikov (1987), and on more recent results of R. Novikov (1998–2019), A. Agaltsov, T. Hohage, R. Novikov (2019). This paper is an extended version of the talk given at the 12th ISAAC Congress, Aveiro, Portugal, 29 July–2 August, 2019.
KW - Helmholtz equation
KW - Inverse scattering
KW - Phase retrieval
KW - Schrödinger equation
KW - Tomography
U2 - 10.1007/978-3-030-97127-4_3
DO - 10.1007/978-3-030-97127-4_3
M3 - Conference contribution
AN - SCOPUS:85130924588
SN - 9783030971267
T3 - Springer Proceedings in Mathematics and Statistics
SP - 75
EP - 98
BT - Mathematical Analysis, its Applications and Computation - ISAAC 2019
A2 - Cerejeiras, Paula
A2 - Reissig, Michael
PB - Springer
T2 - 12th International Society for Analysis, its Applications and Computation, ISAAC 2019
Y2 - 29 July 2019 through 2 August 2019
ER -