Fourier integral operators and the index of symplectomorphisms on manifolds with boundary

Given two compact manifolds with boundary X, Y  , and a boundary preserving symplectomorphism χ:T⁎Y∖0→T⁎X∖0χ:T⁎Y∖0→T⁎X∖0, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with χ. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how – in the spirit of a classical construction by A. Weinstein – a Fredholm operator of this type can be associated with χ   and a section of the Maslov bundle. If dim⁡Y>2dim⁡Y>2 or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.