Julien Sabin (Paris-Orsay)
Spectral cluster bounds for orthonormal functions
Abstract:
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.