  The advent of transformation optics and metamaterials has made possible
devices producing extreme effects on wave propagation. Here we give theoretical
designs for devices, Schr\"odinger hats, acting as invisible concentrators of
waves. These exist for any wave phenomenon modeled by either the Helmholtz or
Schr\"odinger equations, e.g., polarized waves in EM, pressure waves in
acoustics and matter waves in QM, and occupy one part of a parameter space
continuum of wave-manipulating structures which also contains standard
transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM
and acoustic Schr\"odinger hats, the resulting centralized wave is a localized
excitation. In QM, the result is a new charged quasiparticle, a \emph{quasmon},
which causes conditional probabilistic illusions. We discuss possible solid
state implementations.
