Transparent potentials at fixed energy in dimension two. Fixedenergy dispersion relations for the fast decaying potentials
Abstract
For the twodimensional Schrödinger equation $$[  \Delta + v(x)]\psi = E\psi , x \in \mathbb{R}^2 , E = E_{fixed} > 0 (*)$$ at a fixed positive energy with a fast decaying at infinity potentialv(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 classS=C_{∞}^{(∞)}(&R;^{2}). For the potentials with zero scattering amplitude at a fixed energyE_{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 twodimensional sphericallysymmetric real potentials from the Schwartz classS transparent at a given energy. For the twodimensional 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 KdVtype equations in dimension 2+1 related with the scattering problem (*) (the NovikovVeselov 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 1995
 DOI:
 10.1007/BF02099609
 arXiv:
 arXiv:solvint/9410003
 Bibcode:
 1995CMaPh.174..409G
 Keywords:

 Neural Network;
 Initial Data;
 Decay Rate;
 Dispersion Relation;
 Quantum Computing;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 High Energy Physics  Theory;
 Mathematics  Functional Analysis
 EPrint:
 38 pages, TeX