Alix Deleporte: Semiclassical evolution with non-self-adjoint generators

Abstract: 

The celebrated Egorov theorem relates the time propagation of unitary, linear PDE evolution, in a large frequency or small semiclassical parameter limit, with classical dynamics. Its consequences include the validity of geometric optics and Newtonian mechanics at human scale.

In the case of non-unitary propagation, much less is known about the large frequency/semiclassical behaviour or even the presumed "classical" object. The first step in this direction was the work of Hörmander on pseudodifferential operators with quadratic symbols, which has important and modern applications to control theory and hypoellipticity.

In a joint work with Steve Zelditch, we study the "anti-unitary" case \(hu'=\operatorname{Op}(a)u\) where \(\operatorname{Op}(a)\) is a semiclassical self-adjoint operator. We completely describe the time evolution and the associated "classical" object turns out to be the solution of a geodesic equation in a space of Riemannian metrics. A comfortable setting for this description is a generalised wavepacket or FBI point of view, called "Berezin-Toeplitz quantization", and our description holds in short time for general real-analytic symbols.

In this talk I will introduce the relevant objects, and if the time permits, I will state our results.