  In a recent letter [Phys. Rev. Lett. {\bf 100}, 164101 (2008)] and within the
context of quantized chaotic billiards, random plane wave and semiclassical
theoretical approaches were applied to an example of a relatively new class of
statistical measures, i.e. measures involving both complete spatial integration
and energy summation as essential ingredients. A quintessential example comes
from the desire to understand the short-range approximation to the first order
ground state contribution of the residual Coulomb interaction. Billiards, fully
chaotic or otherwise, provide an ideal class of systems on which to focus as
they have proven to be successful in modeling the single particle properties of
a Landau-Fermi liquid in typical mesoscopic systems, i.e. closed or nearly
closed quantum dots. It happens that both theoretical approaches give fully
consistent results for measure averages, but that somewhat surprisingly for
fully chaotic systems the semiclassical theory gives a much improved
approximation for the fluctuations. Comparison of the theories highlights a
couple of key shortcomings inherent in the random plane wave approach. This
paper contains a complete account of the theoretical approaches, elucidates the
two shortcomings of the oft-relied-upon random plane wave approach, and treats
non-fully chaotic systems as well.
