Quadratic Evolution Equations and Fourier Integral Operators in the Complex Domain
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 Schrodinger 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 family of complex linear canonical transformations acting on a suitable exponentially weighted space of entire functions. A recent result due to Coburn-Hitrik-Sjostrand states that any such FIO can be written uniquely in its so-called 'Bergman form.' As I shall explain, when the solution operator for a general quadratic evolution equation is expressed in its Bergman form, previously out of reach questions can be answered.