The metaplectic semigroup with applications to evolution equations generated by non-selfadjoint quadratic Hamiltonians

Abstract In these notes, we review the construction of the metaplectic semigroup and its applications to evolution equations generated by non-selfadjoint quadratic Hamiltonians arising as Weyl quantizations of complex quadratic forms in phase space. After recalling the classical metaplectic theory and its role in the classical–quantum correspondence, together with the basic tools of isotropic pseudodifferential calculus and isotropic microlocal analysis, we present the construction of the metaplectic semigroup through the geometry of positive complex Lagrangian subspaces. Particular attention is devoted to their characterization and correspondence with tempered Gaussian distributions, which constitute the foundation for the construction of the semigroup. This framework provides a natural geometric description of propagators associated with evolution equations generated by Weyl quantizations of complex quadratic forms and reveals the close relationship between their analytical properties and the underlying symplectic geometry. In the last part of these notes, we discuss recent results on the propagation of singularities, as well as the propagation of concentration in terms of Wigner distributions, for the evolution equations in question.