Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms

We begin with a survey of the standard theory of the metaplectic group with some emphasis on the associated notion of Maslov index. We thereafter introduce the Cayley transform for symplectic matrices, which allows us to study in detail the spreading functions of metaplectic operators, and to prove that they are basically quadratic chirps. As a non-trivial application we give new formulae for the fractional Fourier transform in arbitrary dimension. We also study the regularity of the solutions to the Schrödinger equation in the Feichtinger algebra.