Approximations of pseudo-differential flows

Given a classical symbol $M$ of order zero, and associated semiclassical operators ${\rm op}_\varepsilon(M),$ we prove that the flow of ${\rm op}_\varepsilon(M)$ is well approximated, in time $O(|\ln \varepsilon|),$ by a pseudo-differential operator, the symbol of which is the flow $\exp(t M)$ of the symbol $M.$ A similar result holds for non-autonomous equations, associated with time-dependent families of symbols $M(t).$ This result was already used, by the author and co-authors, to give a stability criterion for high-frequency WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here two further applications: sharp semigroup bounds, implying nonlinear instability under the assumption of spectral instability at the symbolic level, and a new proof of sharp G\r{a}rding inequalities.