Experimental investigation of nodal domains in the chaotic microwave rough billiard

Abstract

We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave functions Psi_N of the rough billiard were measured up to the level number N=435. In this way the dependence of the number of nodal domains \aleph_N on the level number $N$ was found. We show that in the limit N->infty a least squares fit of the experimental data reveals the asymptotic number of nodal domains aleph_N/N = 0.058 +- 0.006 that is close to the theoretical prediction aleph_N/N +- 0.062. We also found that the distributions of the areas s of nodal domains and their perimeters l have power behaviors n_s ~ s^{-tau} and n_l ~ l^{-tau'}, where scaling exponents are equal to τ= 1.99 +- 0.14 and τ'=2.13 +- 0.23, respectively. These results are in a good agreement with the predictions of percolation theory. Finally, we demonstrate that for higher level numbers N = 220-435 the signed area distribution oscillates around the theoretical limit Sigma_{A} = 0.0386 N^{-1}.