Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

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.

Figure 1

Variance of ${\mathcal{S}}_i$ for the cardioid billiard with Dirichlet boundary conditions. The solid line is the RPW expression, Eq. (6), the dashed line is the semiclassical prediction, Eq. (14), and the discrete points are for the cardioid billiard (shown in inset).

Figure 2

In the semiclassical sums, the four orbits that, as $x\to 0$, coalesce into the same (nearly) periodic orbit, need to be aggregated. (a),(b) Two (nearly) periodic orbits. (c),(d) Two nonperiodic orbits ($p_x^{\prime }=-p_x)$.