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

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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 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 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.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Kosygina 2, 117940, Moscow, Russia

    Piotr G. Grinevich

  2. CNRS, U.R.A. 758, Département de Mathématiques, Université de Nantes, F-44072, Nantes Cedex 03, France

    Roman G. Novikov

Authors
  1. Piotr G. Grinevich
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  2. Roman G. Novikov
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Correspondence to Piotr G. Grinevich.

Additional information

Communicated by B. Simon

The main part of this work was fulfilled during the visit of one of the authors (P.G.G.) to the University of Nantes in June 1994. He is grateful to the University of Nantes for the invitation and the financial support of this visit. He was also supported by the Soros International Scientific foundation grant MD 8000 and by the Russian Foundation for Fundamental Studies grant 93-011-16087.

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Grinevich, P.G., Novikov, R.G. Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials. Commun.Math. Phys. 174, 409–446 (1995). https://doi.org/10.1007/BF02099609

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  • DOI: https://doi.org/10.1007/BF02099609

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