In mathematical physics, non self-adjoint operators and their associated evolution equations are used to model dissipative phenomena. In this talk, I will present some new results concerning the propagation of global analytic singularities and L^p-bounds for solutions of Schrödinger equations on R^n with non self-adjoint quadratic Hamiltonians. The main idea in the proofs is that, after conjugation by a metaplectic Fourier-Bros-Iagolnitzer (FBI) transform, the solution operator for such a quadratic evolution equation is a Fourier integral operator (FIO) associated to a complex canonical transformation acting on a suitable exponentially weighted space of entire functions. When this FIO is written in its so-called `Bergman form', previously out of reach questions can be answered.
