Speaker: Xianchao WU (Wuhan University of Technology)

Time： June 1, 16:00-17:00

Place： 5406

Title： Distribution of Schr\{o}dinger eigenfunctions

Abstract： In this talk, we consider the distribution of eigenfunctions of the semiclassical Schr\{o}dinger operator on a compact manifold. Their behaviors in forbidden regions and classically allowed regions are dramatically different.

In forbidden regions, we will introduce a partial converse to the Agmon estimates (ie. exponential lower bounds for the eigenfunctions) in terms of Agmon distance under a control assumption. Then by considering a Dirichlet problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we will show a sharp reverse Agmon estimate on a hypersurface in the analytic setting.

However, in classically allowed regions, the behavior of Schr\{o}dinger eigenfunctions is much more complicated. Intuitively, the more time a packet spends near a hypersurface the more concentration we would expect to see there. How to describe it quantitively? We show that if the defect measure $\mu$ associated to a sequence of Schr\{o}dinger eigenfunctions is $\epsilon_0$- tangentially diffuse with respect to the hypersurface, then one can get $o(1)$ improvement of the well known $O(h^{-1/4})$ restriction bounds.