# 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).

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Contributed by: Jessica Alfonsi (Padova, Italy) (December 2020)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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

References

[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.

## Permanent Citation