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Title: $L^p$ bounds on eigenfunctions for operators with double characteristics

Abstract: Starting with the pioneering works of Hormander and Sogge, the question of establishing uniform and, more generally, $L^p$ estimates for eigenfunctions of elliptic self-adjoint operators in the high energy limit has been of fundamental significance in spectral theory and applications. In this talk, after a brief introduction to this circle of questions, we shall discuss $L^p$ bounds on the ground states for a class of semiclassical operators with double characteristics, including some Schrodinger operators with complex-valued potentials. Sharp bounds are obtained under the assumption that the quadratic approximations along the double characteristics are elliptic. This is a joint work with Gunther Uhlmann.