Exact eigenfunction amplitude distributions of integrable quantum billiards

Abstract

The exact probability distributions of the amplitudes of eigenfunctions, $\Psi (x,y)$, of several integrable planar billiards are analytically calculated and shown to possess singularities at $\Psi =0$; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, $\mathbf{K}(1-{\Psi }^2)$, and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about $\Psi =0$.

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Cited by in corpus (4)

• Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems
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• Signatures of Chaos in Non-integrable Models of Quantum Field Theory
• Morphology of three-body quantum states from machine learning