Julien Sabin (Paris-Orsay)

Spectral cluster bounds for orthonormal functions

Sogge's L^p bounds are a way to measure the concentration of eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds associated to large eigenvalues. We generalize these bounds to systems of orthonormal functions, building a bridge between Sogge's result about concentration and the Weyl law, which in some sense is a manifestation of non-concentration. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators. Joint work with Rupert Frank.