Steklov Institute of Mathematics at St.Petersburg

PREPRINT 21/2002

S. A. Avdonin and M. I. Belishev

Dynamical inverse problem for the Schr\"{o}dinger equation (the BC-method)

This preprint was accepted December 3, 2002
Contact: M. I. Belishev

Let $\Omega \subset {\RE}^n$ be a bounded domain; $\partial \Omega
\in C^2$; $\rho, q \in C^2(\overline \Omega)$ real
> 0$. We show that for any fixed $T>0$ the response operator $R^T
:f\to u|_{\partial \Omega \times [0,T]}$ of the Schr\"{o}dinger
system $i\rho\, u_t +\Delta u -qu=0$ in $\Omega \times (0,T),
\quad u|_{t=0}=0,\quad \frac {\partial u} {\partial \nu}
|_{\partial \Omega \times [0,T]}=f$ determines the coefficients
$\rho =\rho (x), q=q(x),\,\,\, x\in \Omega$ uniquely. The problem
is reduced to one of recovering $\rho ,q$ through the boundary
spectral data. The spectral data are extracted from $R^T$ by the
use of a variational principle. A peculiarity of the approach (the
Boundary Control method) is that it allows to solve the problem
using the data on a finite time interval, avoiding a continuation
beyond [0,T].

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