Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

TL;DR

This study experimentally investigates the relationship between nodal domains and level number in a 3D chaotic microwave billiard, confirming theoretical predictions and percolation theory in the asymptotic limit.

Contribution

It introduces an experimental method to analyze nodal domains in 3D microwave billiards using a 2D level number concept, validating theoretical ratios.

Findings

The experimental ratio of nodal domains to level number is approximately 0.059.

Results are consistent with the theoretical prediction of 0.062.

The study confirms the applicability of percolation theory to 3D chaotic systems.

Abstract

We show that using the concept of the two-dimensional level number N_{|} one can experimentally study of the nodal domains in a three-dimensional (3D) microwave chaotic rough billiard with the translational symmetry. Nodal domains are regions where a wave function has a definite sign. We found the dependence of the number of nodal domains aleph_{N_{|}} lying on the cross-sectional planes of the cavity on the two-dimensional level number N_{|}. We demonstrate that in the limit N_{|} -> infinity the least squares fit of the experimental data reveals the asymptotic ratio aleph_{N_{|}}/N_{|} = 0.059 +- 0.029 that is close to the theoretical prediction aleph_{N_{|}}/N_{|} = 0.062. This result is in good agreement with the predictions of percolation theory.