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## N. Anantharaman: Mesures semiclassiques pour l’équation de Schrödinger sur le tore

(draft) Semi-classical analysis deals with high-frequency limits and is the mathematical counterpart of physical questions like: in which sense is classical mechanics the limit of quantum mechanics when $\hbar\to0$? do stationary waves induce measures that converge toward the Liouville measure as the frequency converge to $\infty$?

The celebrated Quantum Unique Ergodicity conjecture of Rudnick and Sarnak (1994) states that this is the case on compact manifolds with strictly negative curvature. E. Lindenstrauss has solved this problem in the so-called arithmetic case.

N. Anantharaman showed that, on a manifold with constant negative curvature, all the accumulation point of the induced measures have Hausdorff dimension at least half the dimension of the cotangent bundle. G. Rivière has recently extended this to variable negative curvature for surfaces.

On the (flat) torus one could think that these questions become trivial. But this is not the case: they involve rather deep arithmetical questions like the number of integer points on arbitrary spheres.

N. Anantharaman and F. Macia have recently proved the following:

Theorem. Let $\mathbb T^d$ be the $d$-dimensional torus. Let $u_n\in L^2(\mathbb T^d)$ with $\|u_n\|_{L^2}=1$. Let $\nu_n$ be the measure defined by: $\nu_n(dx) = \int_0^1 \left|e^{it\Delta/2}u_n\right|^2 dt \cdot dx$. Then any accumulation point of $\nu_n,n\geq1$, is absolutely continuous.

This generalizes (using different techniques) a result of Bourgain and D. Jakobson.