This thesis contains a detailed study of the rates of wave decay for scattering by thin barriers. Thin barriers are systems in which, except for a narrow region, waves do not interact. This type of behavior is observed in physical systems including concert halls and quantum corrals. A quantum corral is constructed by configuring individual atoms or molecules to form a barrier which partially confines electrons to its interior. Here, the atoms produce a potential which plays the role of the thin barrier. In the setting of concert halls, the walls play the role of the barrier and produce partial confinement of sound waves.

Rather than studying systems with a finite width interaction region, we imagine that the interaction region is reduced to a single hypersurface in R^d by taking a limit of barriers whose width is decreasing and intensity is increasing.

Specifically, we are interested in wave equations with delta and delta' potentials on a hypersurface. These operators are used as models for leaky quantum graphs and quantum corrals.