Semiclassical measures for the Schrödinger equation on the torus
Résumé
Our main result is the following : let $(u_{n})$ be a sequence in $L^{2}(\mathbb{T}^{d})$, such that $\norm{u_n}_{L^{2}(\mathbb{T}^{d})}=1$ for all $n$. Consider the sequence of probability measures $\nu_{n}$ on $\mathbb{T}^{d}$, defined by $\nu_{n}(dx)=\left( \int_{0}^{1}|e^{it\Delta /2}u_{n}(x)|^{2}dt\right) dx. $ Let $\nu$ be any weak-$\ast$ limit of the sequence $(\nu_{n})$~: then $\nu$ is absolutely continuous. This generalizes a former result of Bourgain and Jakobson, who considered the case when the functions $u_{n}$ are eigenfunctions of the Laplacian. Our approach is different from theirs, it relies on the notion of (two-microlocal) semiclassical measures, and the properties of the geodesic flow on the torus.
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