Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators

I will present some general tools to factorize semigroups generated by inhomogeneous quadratic differential operators as product of more elementary semigroups. On the one hand, I will explain how such a factorization can be used to characterize the regularizing effects of these semigroups. On the other hand, from a numerical point of view, we will see how it can be used to get some efficient and accurate splitting methods. I will highlight the efficiency of these new methods on the example of the magnetic linear Schrödinger equations with quadratic potentials and their natural nonlinear generalizations. This talk will rely on some joint works with Paul Alphonse, Nicolas Crouseilles, Fernando Casas and Yingzhe Li.