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Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity. (English. Russian original) Zbl 1022.81057
Russ. Math. Surv. 55, No. 6, 1015-1083 (2000); translation from Usp. Mat. Nauk 55, No. 6, 3-70 (2000).
This paper considers the problem of reconstructing the potential of a two dimensional Schrödinger operator from scattering data at fixed energy. This problem is equivalent to the fixed frequency inverse scattering acoustic problem, and hence, the results apply in both cases. As this problem possesses an infinite-dimensional symmetry algebra, generated by the Novikov-Veselov hierarchy, it is in some sense exactly soluble. The author gives a detailed review of results obtained by himself and by other authors. The methods of modern soliton theory are heavily used.

81U40 Inverse scattering problems in quantum theory
35Q51 Soliton equations
35P25 Scattering theory for PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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