Local formula for the index of a Fourier Integral Operator

Abstract

We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $\varphi$ of cosphere bundles of two Riemannian manifolds X and Y is given by ${\int }_{B^{\ast }X}\hat{A}(T^{\ast }X)\exp \theta -{\int }_{B^{\ast }Y}\hat{A}(T^{\ast }Y)\exp \theta$. Here $B^{\ast }$ stands for the unit coball bundle and $\theta$ is a certain characteristic class depending on the principal symbol of the Fourier integral operator. In the special case when X=Y we obtain a different proof of the theorem of Epstein and Melrose.

25 pages, preprint