Partial Differential Operators and Fourier Integral Operators in the Gevrey setting and applications

We state and prove here semiclassical results about the construction of asymptotic solutions by the WKB method for pseudo-differential equations of real principal type. It is a Gevrey version; the smooth $C^{\mathrm{\infty }}$ and the analytic ones may be found in Hörmander (Hormander1985) and Sjöstrand (Sjöstrand1982). We present here three versions depending on the degree of entrance in the complex domain.