Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentialsстатья

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[1] Grinevich P. G., Novikov R. G. Transparent potentials at fixed energy in dimension two. fixed-energy dispersion relations for the fast decaying potentials // Communications in Mathematical Physics. — 1995. — Vol. 174. — P. 409–446. For the two-dimensional Schrodinger equation [-Delta +v(x)]psi=Epsi , xin R2, E =E_{fixed} > 0 at a fixed positive energy with a fast decaying at infinity potential v(x) dispersion relations on the scattering data are given. Under small norm assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization o f scattering data for the potentials from the Schwartz class. For the potentials with zero scattering amplitude at a fixed energy (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parametrized by a function of one variable) of two-dimensional spherically-symmetric real potentials from the Schwartz class transparent at a given energy. For the two-dimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing, at infinity, potentials transparent at a fixed energy. For any dimension greater or equal to 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdV-type equations in dimension 2+1 related with the scattering problem (the Novikov Veselov equations) do not preserve, in general, these dispersion relations starting from the second one. As a corollary these equations do not preserve, in general, the decay rate faster than x{-3} for initial data from the Schwartz class. [ DOI ]

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