  We introduce the notion of Schr\"odinger integral operators and prove sharp
local and global regularity results for these (including propagators for the
quantum mechanical harmonic oscillator). Furthermore we introduce general
classes of oscillatory integral operators with inhomogeneous phase functions,
whose local and global regularity are also established in classical function
spaces (both in the Banach and quasi-Banach scales). The results are then
applied to obtain optimal (local in time) estimates for the solution to the
Cauchy problem for variable-coefficient Schr\"odinger equations as well as
other evolutionary partial differential equations.
