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Communications in Mathematical Physics

, Volume 174, Issue 2, pp 409–446 | Cite as

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

  • Piotr G. Grinevich
  • Roman G. Novikov
Article

Abstract

For the two-dimensional 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 (∞) (ℝ2). For the potentials with zero scattering amplitude at a fixed energyEfixed (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 classS 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.

Keywords

Neural Network Initial Data Decay Rate Dispersion Relation Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.CNRS, U.R.A. 758, Département de MathématiquesUniversité de NantesNantes Cedex 03France

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