EVENT DETAILS AND ABSTRACT

Title: Semiclassical measures for the Schrödinger equation on the torus (joint work with F. Macia)
Speaker: Nalini Anantharaman
Speaker Info: Orsay
Brief Description:
Special Note:
Abstract:

Our main result is the following : let $(u_{n})$ be a sequence in $L^{2}(\mathbb{T}^{d})$, such that $ orm{u_n}_{L^{2}(\mathbb{T}^{d})}=1$ for all $n$. Consider the sequence of probability measures $ u_{n}$ on $\mathbb{T}^{d}$, defined by $ u_{n}(dx)=(\int_{0}^{1}|e^{it\Delta /2}u_{n}(x)|^{2}dt) dx. $ Let $ u$ be any weak limit of the sequence $( u_{n})$: then $ u$ 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.