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For the two-dimensional Schr\"odinger equation $$ [- \Delta+v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixedpositive energy with a fast decaying at infinity potential $v(x)$ dispersionrelations on the scattering data are given.Under "small norm" assumption usingthese dispersion relations we give (without a complete proof of sufficiency) acharacterization of scattering data for the potentials from the Schwartz class$S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zeroscattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (transparentpotentials) we give a complete proof of this characterization. As a consequencewe construct a family (parameterized by a function of one variable) oftwo-dimensional spherically-symmetric real potentials from the Schwartz class$S$ transparent at a given energy. For the two-dimensional case (withoutassumption that the potential is small) we show that there are no nonzero realexponentially decreasing at infinity, potentials transparent at a fixed energy.For any dimension greater or equal 1 we prove that there are no nonzero realpotentials with zero forward scattering amplitude at an energy interval. Weshow that KdV-type equations in dimension 2+1 related with the scatteringproblem $(*)$ (the Novikov-Veselov equations) do not preserve, in general,these dispersion relations starting from the second one. As a corollary theseequations do not preserve, in general , the decay rate faster then $|x|^{-3}$for initial data from the Schwartz class.Comment: 38 pages, Te

Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials