Asymptotic Lower Bounds for a class of Schroedinger Equations

Details Asymptotic Lower Bounds for a class of Schroedinger Equations Asymptotic Lower Bounds for a class of Schroedinger Equations

2007-12-21

arxiv.org/abs/0712.3648v1

Mathematics

Analysis of PDEs

24 pages. to appear on Comm. Math. Phys

Scientific paper

10.1007/s00220-008-0432-6

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.

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