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

Abstract

For the two-dimensional Schrödinger equation $$[-\Delta +v(x)]\psi =E\psi ,x\in {\mathbb{R}}^2,E=E_{fixed}>0(\ast )$$ 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 of scattering data for the potentials from the Schwartz class $S=C_{\infty }^{(\infty )}({\mathbf{R}}^2).$ 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 (parameterized 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 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 $(\ast )$ (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 then $|x{|}^{-3}$ for initial data from the Schwartz class.

38 pages, TeX

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