Random-Matrix Eigenvalue Statistics for Quantum Billiards

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This Demonstration shows an interesting result for the spectral properties of energy levels of quantum billiards. This result is obtained by solving the 2D stationary Schrödinger equation with Dirichlet boundary conditions over a circular or a cardioid-shaped domain using the finite element method as described in the Wolfram Language documentation for the function NDEigensystem [1]. The spectral statistics of eigenvalue spacings are subsequently analyzed with the tools from random matrix theory (RMT).


When the billiard shape is circular, the classical limit for the dynamics is regular (completely integrable with conservation of both energy and momentum) and the set of unfolded eigenvalues follows Poisson statistics. For a cardioid billiard, the level spacing distribution for the energy eigenvalues in a heart-shaped region (where the classical limit is chaotic) is described by the RMT Wigner's surmise function for the Gaussian orthogonal ensemble (GOE) [2, 3].

The controls let you choose between the circular or cardioid domain shapes. You can inspect both the spectral statistics for the level spacings or the density of energy levels, which are used for scaling eigenvalue spacings in the spectral unfolding procedure, as detailed in the code.


Contributed by: Jessica Alfonsi (Padova, Italy) (December 2020)
Open content licensed under CC BY-NC-SA



Snapshot 1: distribution of level spacings in a circle billiard follows Poisson statistics

Snapshot 2: distribution of level spacings in a heart-shaped billiard follows GOE statistics

Snapshot 3: density of energy levels in a circle billiard

Snapshot 4: density of energy levels in a cardioid billiard


[1] Wolfram Research (2015), "NDEigensystem," Wolfram Language & System Documentation Center. (Nov 24, 2020) reference.wolfram.com/language/ref/NDEigensystem.html.

[2] T. Kriecherbauer, J. Marklof and A. Soshnikov, "Random Matrices and Quantum Chaos," Proceedings of the National Academy of Sciences of the United States of America, 98(19), 2001 pp. 10531–10532. doi:10.1073/pnas.191366198.

[3] A. Bäcker, "Eigenfunctions in Chaotic Quantum Systems," habilitation thesis, Dresden, 2007. (Nov 20, 2020) d-nb.info/98965575X/34.

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