Birkhoff normal forms for Fourier integral operators

Abstract. In this work we construct logarithms and Birkhoff normal forms for elliptic Fourier integral operators in the semi-classical limit, under more general assumptions than in a previous work by the first author. The methods are similar but slightly different. 0. Introduction. This work is a continuation of the work [Ia] of the first author. As in that paper we consider the problem of constructing a microlocal logarithm and a normal form of an elliptic semi-classical Fourier integral operator near a fixed point of the corresponding canonical transformation. In [Ia] the canonical transformation was assumed to be of real hyperbolic type and th