Exact eigenfunction amplitude distributions of integrable quantum billiards

TL;DR

This paper analytically derives the probability distributions of eigenfunction amplitudes in integrable quantum billiards, revealing shape-dependent singularities and contrasting with chaotic billiards' Gaussian behavior.

Contribution

It provides exact analytical expressions for eigenfunction amplitude distributions in integrable billiards, including rectangular and triangular shapes, highlighting their unique singularities and universality.

Findings

Distribution for rectangular billiards involves elliptic integrals.

Triangular billiards' distributions are described by Meijer G and hypergeometric functions.

Eigenfunction fluctuations differ markedly from chaotic billiards' Gaussian distributions.

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, ${\bf K}(1 - \Psi ^2)$, 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 $\Psi = 0$.