Nodal portraits of quantum billiards: Domains, lines, and statistics

TL;DR

This paper reviews the properties of nodal domains and lines in quantum billiards, demonstrating their ability to distinguish between different classical dynamics and geometric shapes, with connections to percolation theory and various modeling approaches.

Contribution

It provides a comprehensive comparison of theoretical and experimental results on nodal patterns, highlighting new modeling techniques and analytical methods for quantum billiard systems.

Findings

Nodal statistics differentiate regular and chaotic dynamics.

Random superpositions model chaotic eigenfunctions effectively.

Connections established between nodal patterns and percolation theory.

Abstract

We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches to counting the nodal domains---using trace formulae, graph theory, and difference equations---are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box'" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity---and far-reaching implications---of the problem.