Ivan Moyano

In this talk I will consider the spectral resolution of the   Laplacian operator on a manifold and discuss the question of how   spectral projectors can concentrate on a given subset of the manifold.  
In particular we will consider two cases : compact manifolds with or  without boundary in which the purely discrete spectrum leads to finite  combinations of eigenfunctions and the unbounded case without  boundaries, in which the spectrum contains a continous part. In both  cases we give quantitative estimates for the localisation of the spectral projection in terms of the highest frequnecy involved, which  are essentially optimal. We also try to refine the uncertainty  principle in this situation so as to consider the smallest possible  localisation sets, which can be of not too small Hausdorff content.  This is based on joint work with G. Lebeau (Nice) and N. Burq (Orsay).