Approximation of pseudo-differential flows (Q2806005)

This paper deals with a family \(\text{op}_\varepsilon(M)\), \(\varepsilon>0\), of semiclassical pseudo-differential operators associated with a matrix-valued classical symbol \(M(x,\xi)\in \mathbb{C}^{n\times n}\) of order \(0\). Let \(\exp(t\;\text{op}_\varepsilon(M))\) denote the flow of the ordinary differential equation \(\partial_tu=\text{op}_\varepsilon(M)u\), \(u|_{t=0}= u_0\in L^2\). It is proved here that the above defined flow is well approximated in time \(O(|\ln\varepsilon|)\) by a pseudo-differential operator, the symbol of which is the flow \(\exp(tM)\) of \(M\), namely: \(\exp(t\;\text{op}_\varepsilon(M))\simeq\text{op}_\varepsilon(\exp(tM))\). This result was already used by the author and co-authors to give a stability criterion for high-frequency WKB approximations.NEWLINENEWLINE In the paper under review, two new applications are given: sharp semigroup bounds, implying nonlinear instability under the assumption of spectral instability at the symbolic level (Theorem 3.6) and a new proof of sharp Gårding inequalities (Theorem 4.1).