  We shall study the following initial value problem: \begin{equation}{\bf
i}\partial_t u - \Delta u + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times
{\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range
potential, whose radial derivative satisfies some supplementary assumptions.
More precisely we shall present a family of identities satisfied by the
solutions to the previous Cauchy problem. As a by--product of these identities
we deduce some uniqueness results and a lower bound for the so called local
smoothing which becomes an identity in a precise asymptotic sense.
