In a recent letter [Phys. Rev. Lett. {\bf 100}, 164101 (2008)] and within thecontext of quantized chaotic billiards, random plane wave and semiclassicaltheoretical approaches were applied to an example of a relatively new class ofstatistical measures, i.e. measures involving both complete spatial integrationand energy summation as essential ingredients. A quintessential example comesfrom the desire to understand the short-range approximation to the first orderground state contribution of the residual Coulomb interaction. Billiards, fullychaotic or otherwise, provide an ideal class of systems on which to focus asthey have proven to be successful in modeling the single particle properties ofa Landau-Fermi liquid in typical mesoscopic systems, i.e. closed or nearlyclosed quantum dots. It happens that both theoretical approaches give fullyconsistent results for measure averages, but that somewhat surprisingly forfully chaotic systems the semiclassical theory gives a much improvedapproximation for the fluctuations. Comparison of the theories highlights acouple of key shortcomings inherent in the random plane wave approach. Thispaper contains a complete account of the theoretical approaches, elucidates thetwo shortcomings of the oft-relied-upon random plane wave approach, and treatsnon-fully chaotic systems as well.
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