Properties of nodal domains in a pseudointegrable barrier billiard

We investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard. The number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular. This reduction is explained by the appearance of superscars in the barrier billiard, i.e., regular structures, which are embedded into the underlying chaotic wave functions. Furthermore, the area distribution of nodal domains follows the predictions of a percolation model for chaotic systems.

Figure 1

Examples of the nodal domains in a regular rectangular (a), a chaotic quarter of a stadium (b), and a pseudointegrable barrier billiard (c). The white (black) areas indicate positive (negative) values of real billiard wave functions. All these wave functions have state number $N$ about 400.

Figure 2

Examples of the structure of the nodal domains for measured superscars. Top row: horizontal bouncing ball (BB) and inverted V superscars. Bottom row: Diamond (D) and W superscars. The lines indicate the central periodic orbit of the corresponding family.

Figure 3

Left: Rescaled number of nodal domains as function of the state number for the measured barrier billiard wave functions (dots). Right: The same for RPWM wave functions (dots). The solid curves correspond to the fit of $n\left(N\right)∕N=a+b∕\sqrt{N}$ to the data.

Figure 4

Mean number of nodal domains inside the POC, $a_{\mathrm{POC}}$, as function of $c_{\max }^2$.

Figure 5

Left: Area distribution of nodal domains averaged over all measured wave functions of the barrier billiard on log-log scale. Right: The same averaged over all wave functions for the RPWM. The solid line with slope $181∕97$ shows the prediction of the percolation model [12].