Asympototic Statistics of Nodal Domains in Quantum Chaotic Billiards in the Semiclassical Limit

Kyle Konrad '12

CS-math major, Dartmouth College

Quantum chaos concerns eigenfunctions of the Laplace operator in a domain where a billiard ball would bounce chaotically. Such chaotic eigenfunctions, have been conjectured to share statistical properties of their nodal domains with a simple percolation model, from which many interesting quantities can be computed analytically. We numerically test conjectures on the number and size of nodal domains of quantum chaotic eigenfunctions at very high energies, approaching the semiclassical limit. We use a highly efficient scaling method to quickly compute eigenfunctions at low resolution and interpolate to higher resolution. We collect around 100,000 eigenfunctions and count 1.5 billion nodal domains. Our results provide strong numerical evidence for conjectures on the mean number and size of nodal domains but disagree with the conjectured variance.

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