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Uspekhi Mat. Nauk, 1987, Volume 42, Issue 3(255), Pages 93–152 (Mi umn2537)  

This article is cited in 65 scientific papers (total in 66 papers)

The $\bar\partial$-equation in the multidimensional inverse scattering problem

R. G. Novikov, G. M. Henkin

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English version:
Russian Mathematical Surveys, 1987, 42:3, 109–180

Bibliographic databases:

UDC: 517.9+530.1
MSC: 81U40, 81Q05, 35J10
Received: 23.10.1986

Citation: R. G. Novikov, G. M. Henkin, “The $\bar\partial$-equation in the multidimensional inverse scattering problem”, Uspekhi Mat. Nauk, 42:3(255) (1987), 93–152; Russian Math. Surveys, 42:3 (1987), 109–180

Citation in format AMSBIB
\by R.~G.~Novikov, G.~M.~Henkin
\paper The~$\bar\partial$-equation in the multidimensional inverse scattering problem
\jour Uspekhi Mat. Nauk
\yr 1987
\vol 42
\issue 3(255)
\pages 93--152
\jour Russian Math. Surveys
\yr 1987
\vol 42
\issue 3
\pages 109--180

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    This publication is cited in the following articles:
    1. P. G. Grinevich, S. P. Novikov, “Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I. Energies below the ground state”, Funct. Anal. Appl., 22:1 (1988), 19–27  mathnet  crossref  mathscinet  zmath  isi
    2. R. G. Novikov, “Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$”, Funct. Anal. Appl., 22:4 (1988), 263–272  mathnet  crossref  mathscinet  zmath  isi
    3. R Beals, R R Coifman, Inverse Probl, 5:2 (1989), 87  crossref  mathscinet  zmath  adsnasa  isi
    4. R Weder, Inverse Probl, 7:3 (1991), 461  crossref  mathscinet  zmath  adsnasa  isi
    5. Ricardo Weder, “Generalized limiting absorption method and multidimensional inverse scattering theory”, Math Meth Appl Sci, 14:7 (1991), 509  crossref  mathscinet  zmath  isi
    6. G. Eskin, J. Ralston, “Inverse backscattering in two dimensions”, Comm Math Phys, 138:3 (1991), 451  crossref  mathscinet  zmath  adsnasa  isi
    7. A.G. Ramm, O.L. Weaver, “3D inverse scattering”, Computers & Mathematics with Applications, 22:4-5 (1991), 1  crossref
    8. Masaru Ikehata, “A special green's function for the biharmonic operator and its application to an inverse boundary value problem”, Computers & Mathematics with Applications, 22:4-5 (1991), 53  crossref
    9. David Colton, Lassi Päivärinta, “The uniqueness of a solution to an inverse scattering problem for electromagnetic waves”, Arch Rational Mech Anal, 119:1 (1992), 59  crossref  mathscinet  zmath  isi
    10. G. Eskin, J. Ralston, “Inverse backscattering”, J Anal Math, 58:1 (1992), 177  crossref  mathscinet  zmath  isi
    11. V. V. Dubrovskii, “On stability of inverse problems of spectral analysis for equations of mathematical physics”, Russian Math. Surveys, 49:3 (1994), 183–184  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. Volker Enss, Ricardo Weder, “The geometrical approach to multidimensional inverse scattering”, J Math Phys (N Y ), 36:8 (1995), 3902  crossref  mathscinet  zmath  isi
    13. G Hachem, Inverse Probl, 11:1 (1995), 123  crossref  mathscinet  zmath  adsnasa  isi
    14. G. Eskin, J. Ralston, “Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy”, Comm Math Phys, 173:1 (1995), 199  crossref  mathscinet  zmath  adsnasa  isi
    15. Silke Arians, “Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials”, J Math Phys (N Y ), 38:6 (1997), 2761  crossref  mathscinet  zmath  isi
    16. Gulmaro Corona Corona, “Generalized Gel’fand–Levitan integral equation for two block Ablowitz–Kaup–Newell–Segur systems”, J Math Phys (N Y ), 40:9 (1999), 4393  crossref  mathscinet  zmath  isi
    17. O. M. Kiselev, “Asymptotic behaviour of the solution of the two-dimensional Dirac system with rapidly oscillating coefficients”, Sb. Math., 190:2 (1999), 233–254  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Proc. Steklov Inst. Math., 225 (1999), 285–302  mathnet  mathscinet  zmath
    19. P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. Grant Karamyan, “The inverse scattering problem for the acoustic equation in a half-space”, Inverse Probl, 18:6 (2002), 1673  crossref  mathscinet  zmath  isi  elib
    21. Yu. V. Zasorin, “Potential Renormalization Method for a Model of the Hartree–Fock–Slater Type”, Theoret. and Math. Phys., 130:3 (2002), 375–382  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    22. Grant Karamyan, “Inverse Scattering in a Half Space with Passive Boundary”, Communications in Partial Differential Equations, 28:9-10 (2003), 1627  crossref
    23. Grant Karamyan, “The inverse scattering problem with impedance boundary in a half-space”, Inverse Probl, 20:5 (2004), 1485  crossref  mathscinet  zmath  isi  elib
    24. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    25. Alexandru Tamasan, “On the scattering method for the -equation and reconstruction of convection coefficients”, Inverse Problems, 20:6 (2004), 1807  crossref
    26. Ricardo Weder, Dimitri Yafaev, “On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity”, Inverse Probl, 21:6 (2005), 1937  crossref  mathscinet  zmath  isi
    27. R G Novikov, “Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential”, Inverse Probl, 21:1 (2005), 257  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. Xiaosheng Li, “Inverse Scattering Problem for the Schrödinger Operator with External Yang–Mills Potentials in Two Dimensions at Fixed Energy”, Communications in Partial Differential Equations, 30:4 (2005), 451  crossref
    29. David Dos Santos Ferreira, Carlos E. Kenig, Johannes Sjöstrand, Gunther Uhlmann, “Determining a Magnetic Schrödinger Operator from Partial Cauchy Data”, Comm Math Phys, 271:2 (2007), 467  crossref  mathscinet  zmath  isi
    30. A S Fokas, “Nonlinear Fourier transforms, integrability and nonlocality in multidimensions”, Nonlinearity, 20:9 (2007), 2093  crossref  mathscinet  zmath  adsnasa  isi
    31. Mikko Salo, “Recovering first order terms from boundary measurements”, J Phys Conf Ser, 73 (2007), 012020  crossref
    32. I. A. Taimanov, S. P. Tsarev, “Two-dimensional Schrödinger operators with fast decaying potential and multidimensional $L_2$-kernel”, Russian Math. Surveys, 62:3 (2007), 631–633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    33. Isozaki, H, “The partial derivative-theory for inverse problems associated with Schrodinger operators on hyperbolic spaces”, Publications of the Research Institute For Mathematical Sciences, 43:1 (2007), 201  crossref  mathscinet  zmath  isi
    34. I. A. Taimanov, S. P. Tsarev, “Two-dimensional rational solitons and their blowup via the Moutard transformation”, Theoret. and Math. Phys., 157:2 (2008), 1525–1541  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    35. Shari Moskow, John C Schotland, “Convergence and stability of the inverse scattering series for diffuse waves”, Inverse Problems, 24:6 (2008), 065005  crossref
    36. Burov V.A., Alekseenko N.V., Rumyantseva O.D., “Mnogochastotnoe obobschenie algoritma Novikova dlya resheniya obratnoi dvumernoi zadachi rasseyaniya”, Akusticheskii zhurn., 55:6 (2009), 784–798  elib
    37. G Uhlmann, “Electrical impedance tomography and Calderón's problem”, Inverse Problems, 25:12 (2009), 123011  crossref
    38. I. A. Taimanov, S. P. Tsarev, “On the Moutard transformation and its applications to spectral theory and soliton equations”, Journal of Mathematical Sciences, 170:3 (2010), 371–387  mathnet  crossref  mathscinet
    39. Lassi Päivärinta, Mikko Salo, Gunther Uhlmann, “Inverse scattering for the magnetic Schrödinger operator”, Journal of Functional Analysis, 259:7 (2010), 1771  crossref
    40. R G Novikov, “New global stability estimates for the Gel'fand–Calderon inverse problem”, Inverse Probl, 27:1 (2011), 015001  crossref
    41. Hiroshi Isozaki, Evgeny Korotyaev, “Inverse Problems, Trace Formulae for Discrete Schrödinger Operators”, Ann. Henri Poincaré, 2011  crossref
    42. Kenig C.E., Salo M., Uhlmann G., “Reconstructions From Boundary Measurements on Admissible Manifolds”, Inverse Probl. Imaging, 5:4 (2011), 859–877  crossref  isi
    43. Burov V.A., Kasatkina E.E., Poberezhskaya A.Yu., Bogatyrev A.V., Rumyantseva O.D., “Osobennosti rascheta protsessov rasseyaniya na kontrastnykh i silno pogloschayuschikh dvukh- i trekhmernykh neodnorodnostyakh”, Akusticheskii zhurnal, 57:5 (2011), 665–680  elib
    44. Matteo Santacesaria, “New global stability estimates for the Calderón problem in two dimensions”, J. Inst. Math. Jussieu, 2012, 1  crossref
    45. Grinevich P.G. Novikov R.G., “Faddeev Eigenfunctions for Point Potentials in Two Dimensions”, Phys. Lett. A, 376:12-13 (2012), 1102–1106  crossref  isi
    46. R. G. Novikov, M. Santacesaria, “Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems”, International Mathematics Research Notices, 2012  crossref
    47. Bazulin E.G., “O vozmozhnosti ispolzovaniya v ultrazvukovom nerazrushayuschem kontrole metoda maksimalnoi entropii dlya polucheniya izobrazheniya rasseivatelei po naboru ekhosignalov”, Akusticheskii zhurnal, 59:2 (2013), 235–235  elib
    48. M. I. Isaev, R. G. Novikov, “Stability estimates for recovering the potential by the impedance boundary map”, St. Petersburg Math. J., 25:1 (2014), 23–41  mathnet  crossref  mathscinet  zmath  isi  elib
    49. V. A. Burov, A. S. Shurup, D. I. Zotov, O. D. Rumyantseva, “Simulation of a functional solution to the acoustic tomography problem for data from quasi-point transducers”, Acoust. Phys, 59:3 (2013), 345  crossref
    50. I. A. Taimanov, S. P. Tsarev, “Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation”, Theoret. and Math. Phys., 176:3 (2013), 1176–1183  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    51. M. I. Isaev, “Exponential Instability in the Inverse Scattering Problem on the Energy Interval”, Funct. Anal. Appl., 47:3 (2013), 187–194  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    52. Novikov R.G., “Approximate Lipschitz Stability for Non-Overdetermined Inverse Scattering at Fixed Energy”, J. Inverse Ill-Posed Probl., 21:6 (2013), 813–823  crossref  isi
    53. A. V. Kazeykina, “Absence of Solitons with Sufficient Algebraic Localization for the Novikov–Veselov Equation at Nonzero Energy”, Funct. Anal. Appl., 48:1 (2014), 24–35  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    54. Gunther Uhlmann, “Inverse problems: seeing the unseen”, Bull. Math. Sci, 2014  crossref
    55. M.I. Isaev, R.G. Novikov, “Effectivized Hölder-logarithmic stability estimates for the Gel’fand inverse problem”, Inverse Problems, 30:9 (2014), 095006  crossref
    56. A. D. Agaltsov, R. G. Novikov, “Riemann–Hilbert problem approach for two-dimensional flow inverse scatteringa)”, J. Math. Phys, 55:10 (2014), 103502  crossref
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    58. V. A. Burov, D. I. Zotov, O. D. Rumyantseva, “Reconstruction of the sound velocity and absorption spatial distributions in soft biological tissue phantoms from experimental ultrasound tomography data”, Acoust. Phys, 61:2 (2015), 231  crossref
    59. Novikov R.G., “Formulas For Phase Recovering From Phaseless Scattering Data At Fixed Frequency”, 139, no. 8, 2015, 923–936  crossref  isi
    60. Novikov R.G., “Explicit Formulas and Global Uniqueness for Phaseless Inverse Scattering in Multidimensions”, J. Geom. Anal., 26:1 (2016), 346–359  crossref  mathscinet  zmath  isi  elib  scopus
    61. Grinevich P.G. Novikov R.G., “Moutard transform approach to generalized analytic functions with contour poles”, Bull. Sci. Math., 140:6 (2016), 638–656  crossref  mathscinet  zmath  isi  elib  scopus
    62. Ferreira David Dos Santos, Kurylev Ya., Lassas M., Salo M., “The Calderón problem in transversally anisotropic geometries”, J. Eur. Math. Soc., 18:11 (2016), 2579–2626  crossref  mathscinet  zmath  isi  scopus
    63. E. L. Lakshtanov, B. R. Vainberg, “A test for the existence of exceptional points in the Faddeev scattering problem”, Theoret. and Math. Phys., 190:1 (2017), 77–90  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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