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.
Categories: math.AP