Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-10-26
Physics
High Energy Physics
High Energy Physics - Theory
38 pages, TeX
Scientific paper
10.1007/BF02099609
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.
Grinevich Piotr G.
Novikov Roman G.
No associations
LandOfFree
Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-314692