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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 6(336), Pages 3–70 (Mi umn333)  

This article is cited in 48 scientific papers (total in 48 papers)

Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity

P. G. Grinevich

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We study the problem of reconstructing the potential of the two-dimensional Schrödinger operator from scattering data measured at fixed energy. This problem, in contrast to the general multidimensional inverse problem, possesses an infinite-dimensional symmetry algebra generated by the Novikov–Veselov hierarchy and hence is “exactly soluble” in some sense; the complexity of the answer is approximately the same as in the one-dimensional problem. We make heavy use of methods developed in modern soliton theory. Since the quantum fixed-energy scattering problem is mathematically equivalent to the acoustic single-frequency scattering problem, we see that the results of the present paper apply in both cases.

DOI: https://doi.org/10.4213/rm333

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English version:
Russian Mathematical Surveys, 2000, 55:6, 1015–1083

Bibliographic databases:

UDC: 517.958+517.984.54
MSC: Primary 81U40; Secondary 35J10, 37K40, 37K15, 35P25, 34L40, 34L20, 35Q53
Received: 31.05.2000

Citation: P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Uspekhi Mat. Nauk, 55:6(336) (2000), 3–70; Russian Math. Surveys, 55:6 (2000), 1015–1083

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    This publication is cited in the following articles:
    1. R G Novikov, “On the range characterization for the two-dimensional attenuated x-ray transformation”, Inverse Probl, 18:3 (2002), 677  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Dubrovsky V.G., Formusatik I.B., “New lumps of Veselov-Novikov integrable nonlinear equation and new exact rational potentials of two-dimensional stationary Schrtsdinger equation via $\overline\partial$-dressing method”, Phys. Lett. A, 313:1-2 (2003), 68–76  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Dubrovsky V.G., Formusatik I.B., “New rational solutions of Veselov-Novikov equation and new exact rational potentials of two-dimensional stationary Schrodinger equation via partial derivative-dressing method”, Korus 2005, Proceedings, 2005, 136–138  isi
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    5. Lassas, M, “Mapping properties of the nonlinear Fourier transform in dimension two”, Communications in Partial Differential Equations, 32:4 (2007), 591  crossref  mathscinet  zmath  isi  scopus  scopus
    6. 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  scopus  scopus
    7. Shan, H, “A globally accelerated numerical method for optical tomography with continuous wave source”, Journal of Inverse and Ill-Posed Problems, 16:8 (2008), 763  crossref  mathscinet  zmath  isi
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    9. V. G. Dubrovsky, A. V. Topovsky, M. Yu. Basalaev, “Two-dimensional stationary Schrödinger equation via the ∂-dressing method: New exactly solvable potentials, wave functions, and their physical interpretation”, J Math Phys (N Y ), 51:9 (2010), 092106  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
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    11. V. G. Dubrovskii, A. V. Topovsky, M. Yu. Basalaev, “New exact solutions with functional parameters of the Nizhnik–Veselov–Novikov equation with constant asymptotic values at infinity”, Theoret. and Math. Phys., 165:2 (2010), 1470–1489  mathnet  crossref  crossref  isi
    12. Klibanov M.V., Su J., Pantong N., Shan H., Liu H., “A globally convergent numerical method for an inverse elliptic problem of optical tomography”, Applicable Analysis, 89:6 (2010), 861–891  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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    17. Novikov R.G., “Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy”, Phys Lett A, 375:9 (2011), 1233–1235  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    18. A V Kazeykina, “A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data”, Inverse Problems, 28:5 (2012), 055017  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    19. M. Lassas, J.L. Mueller, S. Siltanen, A. Stahel, “The Novikov–Veselov equation and the inverse scattering method, Part I: Analysis”, Physica D: Nonlinear Phenomena, 2012  crossref  mathscinet  isi  scopus  scopus
    20. Grinevich P.G., Novikov R.G., “Faddeev Eigenfunctions for Point Potentials in Two Dimensions”, Phys. Lett. A, 376:12-13 (2012), 1102–1106  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    21. V. G. Dubrovsky, A. V. Topovsky, “About simple nonlinear and linear superpositions of special exact solutions of Veselov-Novikov equation”, J. Math. Phys, 54:3 (2013), 033509  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    22. Kudryavtsev A.G., “Exactly Solvable Two-Dimensional Stationary Schrodinger Operators Obtained by the Nonlocal Darboux Transformation”, Phys. Lett. A, 377:38 (2013), 2477–2480  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    23. Santacesaria M., “Stability Estimates for an Inverse Problem for the Schrodinger Equation at Negative Energy in Two Dimensions”, Appl. Anal., 92:8 (2013), 1666–1681  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    24. Watanabe M., “Inverse Scattering for the Stationary Wave Equation with a Friction Term in Two Dimensions”, Publ. Res. Inst. Math. Sci., 49:1 (2013), 155–176  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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    27. V.G. Dubrovsky, A.V. Topovsky, “About linear superpositions of special exact solutions of Nizhnik-Veselov-Novikov equation”, J. Phys.: Conf. Ser, 482 (2014), 012011  crossref  adsnasa  isi  scopus  scopus
    28. 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
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    30. Perry P.A., “Miura Maps and Inverse Scattering For the Novikov-Veselov Equation”, Anal. PDE, 7:2 (2014), 311–343  crossref  mathscinet  zmath  isi  scopus  scopus
    31. R. G. Novikov, “An iterative approach to non-overdetermined inverse scattering at fixed energy”, Sb. Math., 206:1 (2015), 120–134  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    35. Croke R., Mueller J.L., Music M., Perry P., Siltanen S., Stahel A., “the Novikov-Veselov Equation: Theory and Computation”, Nonlinear Wave Equations: Analytic and Computational Techniques, Contemporary Mathematics, 635, eds. Curtis C., Dzhamay A., Hereman W., Prinari B., Amer Mathematical Soc, 2015, 25–70  crossref  mathscinet  zmath  isi
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    39. de Hoop M.V., Lassas M., Santacesaria M., Siltanen S., Tamminen J.P., “Positive-energy D-bar method for acoustic tomography: a computational study”, Inverse Probl., 32:2 (2016), 025003  crossref  mathscinet  zmath  isi  scopus
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