Abstract
The exact probability distributions of the amplitudes of eigenfunctions, Ψ(x, y), of several integrable planar billiards are analytically calculated and shown to possess singularities at Ψ = 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, K(1 − Ψ2), 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 Ψ = 0.
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