Geometric characterization of nodal domains: the area-to-perimeter ratio

Elon, Yehonatan; Gnutzmann, Sven; Joas, Christian; Smilansky, Uzy

doi:10.1088/1751-8113/40/11/007

Cited by 13 publications

(14 citation statements)

References 23 publications

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“…Gaussian waves on R n were first suggested in [2] as a model for the limiting behavior of eigenfunctions of chaotic systems. While the model is not supported by any rigorous derivation, it was found consistent with some numerical observations, such as [7,8,9].…”

Section: Relations With Previous Resultssupporting

confidence: 59%

Gaussian Waves on the Regular Tree

Elon

2009

Preprint

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We consider the family of real (generalized) eigenfunctions of the adjacency operator on T d -the d-regular tree. We show the existence of a unique invariant Gaussian process on the ensemble and derive explicitly its covariance operator.We investigate the typical structure of level sets of the process. In particular we show that the entropic repulsion of the level sets is uniformly bounded and prove the existence of a critical threshold, above which the level sets are all of finite cardinality and below it an infinite component appears almost surely.

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Section: Relations With Previous Resultssupporting

confidence: 59%

Gaussian Waves on the Regular Tree

Elon

2009

Preprint

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“…Throughout this paper if Ω is a Neumann domain of an eigenfunction f of eigenvalue k 2 we will write N (Ω) instead of N (Ω, k). Another parameter that relates the spectrum and geometry of a domain is the rescaled area to perimeter ratio ρ which was introduced in [24], and was used in [10]…”

Section: Neumann Domains and Neumann Countmentioning

confidence: 99%

Neumann Domains on Quantum Graphs

Alon,

Band

2019

Preprint

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The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros of f . The Neumann domains of f are the subgraphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains.We prove some foundational results on Neumann domains of quantum graph eigenfunctions: bounds on the number of Neumann domains and properties of the probability distributions of these numbers. We present fundamental geometric and spectral parameters of Neumann domains, such as the normalized isoperimetric ratio and the spectral position and prove the relevant bounds and properties of the probability distributions.

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“…All of these properties, including the support, are believed to be universal features for all two-dimensional separable surfaces. Explicit derivations of the limiting distributions for the family of simple surfaces of revolution and the disc billiard (Elon et al, 2007) further lend weight to this hypothesis. However, for integrable but non-separable and pseudointegrable billiards, the form of P (ρ) is unknown.…”

Section: Geometric Characterization Of Nodal Domainsmentioning

confidence: 93%

“…The ratio of these two quantities turns out to be another statistically significant tool to sniff out the underlying dynamics of the system. To interrogate the morphology of the nodal lines, Elon et al (2007) considered the set of nodal domains of the j th eigenfunction of a billiard in a domain D; this can be represented as the sequence {ω (m) j }, m = 1, 2, . .…”

Section: Geometric Characterization Of Nodal Domainsmentioning

confidence: 99%

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

Jain,

Samajdar

2017

Preprint

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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

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Geometric characterization of nodal domains: the area-to-perimeter ratio

Cited by 13 publications

References 23 publications

Gaussian Waves on the Regular Tree

Gaussian Waves on the Regular Tree

Neumann Domains on Quantum Graphs

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

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