  We study the effect of edge diffraction on the semiclassical analysis of two
dimensional quantum systems by deriving a trace formula which incorporates
paths hitting any number of vertices embedded in an arbitrary potential. This
formula is used to study the cardioid billiard, which has a single vertex. The
formula works well for most of the short orbits we analyzed but fails for a few
diffractive orbits due to a breakdown in the formalism for certain geometries.
We extend the symbolic dynamics to account for diffractive orbits and use it to
show that in the presence of parity symmetry the trace formula decomposes in an
elegant manner such that for the cardioid billiard the diffractive orbits have
no effect on the odd spectrum. Including diffractive orbits helps resolve peaks
in the density of even states but does not appear to affect their positions. An
analysis of the level statistics shows no significant difference between
spectra with and without diffraction.
