Inverse-Scattering with the time-dependent and time-independent Schroedinger equation
by
Swanhild Bernstein
Bauhaus-Universitaet Weimar, Institute of Mathematics and Physics, 99421 Weimar, Germany

We discuss I.S.T. (Inverse Scattering Transform) schemes for multidimensional equations. The application of the [`(\partial)] method to study nonlinear evolution equations in two spatial dimensions has led to the development of a general formalism to implement inverse scattering in n spatial dimensions by M.J. Ablowitz and A.I. Nachman.
We discuss inverse scattering associated with the Schrödinger-type equation

\sigma  \partialv \partialy + n å j=1   \partial2 v \partialxj2 +u( --> x   , y)v = 0,

with \sigma = \sigma_r+i\sigma_I,  [x\vec]=(x₁, x₂, ... , x_n) in Rⁿ,  y in R. It turns out that there is a very important constraint which the scattering data must satisfy. The inverse scattering problem consists in determining the potential u([x\vec], y) from the inverse data T([k\vec]_R, [k\vec]_I, [(\xi)\vec]). However, it is immediatly obvious that there is a serious redundancy question. Namely T([k\vec]_R, [k\vec]_I, [(\xi)\vec]) is a function of 3n parameters with one restriction, i.e. T will be given as a function of 3n-1 variables and we widh to construct a function u([x\vec], y) depending on n+1 variables. There are several possible reconstruction formulae for u([x\vec], y). However, crucial restrictions on T are imposed by the requirement that u depends only on [x\vec],  y and decays at \infty. This is part of the characterization question: i.e. which inverse data T([k\vec]_R, [k\vec]_I, [(\xi)\vec]) are ädmissible". We will use a Borel-Pompeiu formula in Cⁿ to construct admissible data and use the fact that holomorphic functions are complex monogenic functions. In this way we will need no so-called "miracle condition" or any similar requirement on the inverse data T([k\vec]_R, [k\vec]_I, [(\xi)\vec]).
We will apply our method to the time-dependent and the time-independent Schrödinger equation which are special cases of the general one.

Date received: July 4, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-54.