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Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentialsстатья
Информация о цитировании статьи получена из
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Web of Science
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 22 декабря 2014 г.
- Авторы: Grinevich P.G., Novikov R.G.
- Журнал: Communications in Mathematical Physics
- Том: 174
- Год издания: 1995
- Издательство: Springer Verlag
- Местоположение издательства: Germany
- Первая страница: 409
- Последняя страница: 446
- DOI: 10.1007/BF02099609
- Аннотация: For the two-dimensional Schrodinger 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 o f scattering data for the potentials from the Schwartz class. For the potentials with zero scattering amplitude at a fixed energy (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 class 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.
- Добавил в систему: Гриневич Петр Георгиевич