Defect measures have successfully been used, in a variety of
contexts, as a tool to quantify the lack of compactness of bounded
sequences of square-integrable functions due to concentration and
oscillation effects. In this talk we shall present some results on the
structure of the set of possible defect measures arising from sequences
of solutions to the linear Schrödinger equation on a compact manifold.
This is motivated by questions related to understanding the effect of
geometry on dynamical aspects of the Schrödinger flow, such as
dispersive effects and unique continuation.
It turns out that the answer to these questions depends strongly on
global properties of the geodesic flow on the manifold under
consideration: this will be illustrated by discussing with a certain
detail the examples of the the sphere and the (flat) torus.