TRANSPARENT POTENTIALS AT FIXED-ENERGY IN DIMENSION 2 - FIXED-ENERGY DISPERSION-RELATIONS FOR THE FAST DECAYING POTENTIALS

For the two-dimensional Schrodinger equation [-Delta + upsilon(x)]psi = E psi, x is an element of R(2), E = E(fixed) > 0 (*) at a fixed positive energy with a fast decaying at infinity potential upsilon(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 of scattering data for the potentials from the Schwartz class S = C infinity((infinity)))(IR(2)). For the potentials with zero scattering amplitude at a fixed energy E(fixed) (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 S 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.