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

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. CONTENTS3. Periodic orbits of nonseparable, integrable billiards 48 C. Graph-theoretic analysis 49 D. Difference-equation formalism 50 1. Circles (and annuli/sectors thereof) 50 2. Ellipses and elliptic annuli 50 3. Confocal parabolae 51 4. Right-angled isosceles triangle 51 5. Equilateral triangle 51 E. Counting with Potts spins 52 VII. Experimental realizations 52 A. Microwave billiards: The physicist's pool table 52 B. The S-matrix and transmission measurements 54 C. The perturbing bead method 56 D. Current and vortex statistics 57 VIII. Concluding remarks 60 Acknowledgments 61 A. Isoperimetric inequalities 61 1. Rayleigh-Faber-Krahn and related inequalities 61 2. Payne-Pólya-Weinberger inequality 62 3. Szegö-Weinberger inequality 62 References 62