KTH/SU Mathematics
Colloquium
November 30,
2005
Anders Mellin,
LTH
The backscattering problem in
quantum mechanics
ABSTRACT
The
back-scattering data of the Schr\"odinger operator, may be viewed
as
an 'echo' of the potential. Although the quantum mechanical
representation
of these data are quite complicated there is
alternative way of presentation
that uses simple properties of the
wave equation
$$
(\Delta _x -\p _t ^2 +v)u=f.
$$
This may be solved
by a simple iteration technique which will
allow us to express the
backscattering data as
a function
$$
Bv= \sum _1 ^\infty
B_N(v),
$$
where the $B_N$ are $N$-linear mappings
>From $(C^\infty
(\RRn ))^N $ to $C_0^\infty (\RRn)$ and the series, considered as a
power series in $v$ has infinite radius of convergence.
Each $B_N$
may be viewed as a singular integral operator, and the analysis
of
the continuity properties involves techniques from several areas
of analysis.
In the case $n=1$ the operator $v \to B(v)$ has the
important property that
it linearizes the Korteweg-de Vries equation.
I will also discuss some aspect
of the inversion problem, i.e. the
recovery of $v$ from $B(v)$.