We will consider the space-time distributions associated with solutions of the Schrodinger equation, $\phi(t, x)=e^{it\Delta}\phi_0(x)$.
We will work in the semiclassical limit, i.e. the case when $\phi_0$ has high frequency of oscillation. We will be interested in the measures $dt |\phi(t,x)|^2 dx$.
One can generalize the questions of quantum chaos to this context. In the case of negatively curved manifolds, we will show an entropy result similar to the one presented by Stephane Nonnenmacher for the stationary Schroedinger equation. In addition, we will argue that, for 'almost every' choice of the initial datum $\phi_0$, the measures $dt |\phi(t,x)|^2 dx$ converge to the uniform measure $dt dx$ in the high-frequency limit.
work in progress with Gabriel Riviere
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