Nodal Domains in Chaotic Microwave Rough Billiards with and without Ray-Splitting Properties

We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Breit–Wigner ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. Using the rough billiard without ray-splitting properties we also study the wave functions lying in the regime of Shnirelman ergodicity. The wave functions ΨN of the ray-splitting billiard were measured up to the level number N = 204. In the case of the rough billiard without ray-splitting properties, the wave functions were measured up to N = 435. We show that in the regime of Breit–Wigner ergodicity most of wave functions are delocalized in the n, l basis. In the regime of Shnirelman ergodicity wave functions are homogeneously distributed over the whole energy surface. For such wave functions, lying both in the regimes of Breit–Wigner and Shnirelman ergodicity, the dependence of the number of nodal domains אN on the level number N was found. We show that in the regimes of Breit–Wigner and Shnirelman ergodicity least squares fits of the experimental data reveal the numbers of nodal domains that in the asymptotic limit N →∞ coincide within the error limits with the theoretical prediction אN/N ' 0.062. Finally, we demonstrate that the signed area distribution ΣA can be used as a useful criterion of quantum chaos.