dorsal/arxiv
View SchemaTransparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials
| Authors | Piotr G. Grinevich, Roman G. Novikov |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9410003 |
| URL | https://arxiv.org/abs/solv-int/9410003 |
| DOI | 10.1007/BF02099609 |
Abstract
For the two-dimensional Schr\"odinger equation $$ [- \Delta +v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ 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 $S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zero scattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (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 $(*)$ (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.
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"date_created": "2026-03-02T18:02:48.371000Z",
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"abstract": "For the two-dimensional Schr\\\"odinger equation $$ [- \\Delta\n+v(x)]\\psi=E\\psi,\\ x\\in \\R^2,\\ E=E_{fixed}\u003e0 \\ \\ \\ \\ \\ (*)$$ at a fixed\npositive energy with a fast decaying at infinity potential $v(x)$ dispersion\nrelations on the scattering data are given.Under \"small norm\" assumption using\nthese dispersion relations we give (without a complete proof of sufficiency) a\ncharacterization of scattering data for the potentials from the Schwartz class\n$S=C_{\\infty}^{(\\infty)} (\\hbox{\\bf R}^2).$ For the potentials with zero\nscattering amplitude at a fixed energy $\\scriptstyle E_{fixed}$ (transparent\npotentials) we give a complete proof of this characterization. As a consequence\nwe construct a family (parameterized by a function of one variable) of\ntwo-dimensional spherically-symmetric real potentials from the Schwartz class\n$S$ transparent at a given energy. For the two-dimensional case (without\nassumption that the potential is small) we show that there are no nonzero real\nexponentially decreasing at infinity, potentials transparent at a fixed energy.\nFor any dimension greater or equal 1 we prove that there are no nonzero real\npotentials with zero forward scattering amplitude at an energy interval. We\nshow that KdV-type equations in dimension 2+1 related with the scattering\nproblem $(*)$ (the Novikov-Veselov equations) do not preserve, in general,\nthese dispersion relations starting from the second one. As a corollary these\nequations do not preserve, in general , the decay rate faster then $|x|^{-3}$\nfor initial data from the Schwartz class.",
"arxiv_id": "solv-int/9410003",
"authors": [
"Piotr G. Grinevich",
"Roman G. Novikov"
],
"categories": [
"solv-int",
"funct-an",
"hep-th",
"math.FA",
"nlin.SI"
],
"doi": "10.1007/BF02099609",
"title": "Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials",
"url": "https://arxiv.org/abs/solv-int/9410003"
},
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