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

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

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

Citation in format AMSBIB
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• https://doi.org/10.4213/rm333
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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