Quantum propagation for Berezin–Toeplitz operators

Abstract

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin–Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducing quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.