Enss V.
A New Look at the Multidimensional Inverse Scattering Problem
(51K, LaTeX 2e)
ABSTRACT. As a prototype of an evolution equation we consider the
Schr\"odinger equation $ i (d/dt) \Psi(t) = H \Psi(t), H = H_0 + V(x) $
for the Hilbert space valued function $\Psi(.)$ which describes the
state of the system at time *t* in space dimension at least 2.
The kinetic energy operator $ H_0 $ may be propotional to the
Laplacian (nonrelativistic quantum mechanics),
$ H_0 = \sqrt{-\Delta + m^2} $ (relativistic kinematics, Klein-Gordon
equation), the Dirac operator, or ..., while the potential V(x)
tends to 0 suitably as |x| to infinity.
We present a geometrical approach to the inverse scattering
problem. For given scattering operator S we show uniqueness of
the potential, we give explicit limits of the high-energy behavior of
the scattering operator, and we give reconstruction formulas for the
potential.
Our mathematical proofs closely follow physical intuition. A key
observation is that at high energies translation of wave packets
dominates over spreading during the interaction time.
Extensions of the method cover e.g. Schr\"odinger operators with
magnetic fields, multiparticle systems, and wave equations.
Available through http://www.iram.rwth-aachen.de/~enss/ or
anonymous ftp from work1.iram.rwth-aachen.de (134.130.161.65)
in the directory /pub/papers/enss/ as en-98-1.tex, dvi, ps