Semiclassical construction of random wave functions for confined systems

We develop a statistical description of chaotic wave functions in closed systems obeying arbitrary boundary conditions by combining a semiclassical expression for the spatial two-point correlation function with a treatment of eigenfunctions as Gaussian random fields. Thereby we generalize Berry’s isotropic random wave model by incorporating confinement effects through classical paths reflected at the boundaries. Our approach allows one to explicitly calculate highly nontrivial statistics, such as intensity distributions, in terms of usually few short orbits, depending on the energy window considered. We compare with numerical quantum results for the Africa billiard and derive nonisotropic random wave models for other prominent confinement geometries.

Figure 1

Integrated distribution $P\left(w\right)=\int I(w;\vec{r})d\vec{r}$. The symbols are exact quantum results for the kicked rotor using Eq. (1), the dashed line is the Porter-Thomas distribution $P^{\text{RMT}}\left(w\right)$ and the solid line is $\int I^G(w;\vec{r})d\vec{r}$ with $I^G(w,\vec{r})$ from Eq. (4) and position dependent correlation (3) obtained from quantum mechanical numerical results for the wave functions. The local Gaussian assumption adequately describes the bulk (inset) and tails beyond the RMT result.

Figure 2

Two-point correlation function $R(\vec{r},\vec{r})$ and $R(0,\vec{r})$ (right inset) for $\vec{r}$ pointing along the line indicated in the Africa billiard (left inset). The symbols mark numerical quantum results for $R$, Eq. (3) [30]. The thin lines depict the semiquantum prediction employing Eq. (6) where the Green function is approximated by a sum over paths, including diffraction effects, with at most one reflection at the boundary. The dashed lines show the isotropic RWM result (7).