Exact eigenfunction amplitude distributions of integrable quantum billiards

The exact probability distributions of the amplitudes of eigenfunctions, $\mathrm{\Psi }(x,y)$, of several integrable planar billiards are analytically calculated and shown to possess singularities at $\mathrm{\Psi }=0$; 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, $\mathbf{K}(1-{\mathrm{\Psi }}^2)$, 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 $\mathrm{\Psi }=0$.