New fluctuation properties arise in problems where both spatial integration and energy summation are necessary ingredients. The quintessential example is given by the short-range approximation to the first order ground state contribution of the residual Coulomb interaction. The dominant features come from the region near the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates. Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as compared to Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly to significant errors, which can be corrected within a purely semiclassical approach.

Friedel oscillations of electron densities near impurities have an analog in microwave billiards, which we measure for pseudo-integrable and mixed dynamics geometries. It is expected and found that the oscillations are independent of the dynamics and predicted by a random plane wave model. Separating the chaotic from the regular states for the mixed system requires incorporating the appropriate phase space projection into the modeling in multiple ways for good agreement.