Localization in infinite billiards: a comparison between quantum and classical ergodicity

TL;DR

This paper compares quantum and classical ergodic behavior in a specific non-compact billiard system, revealing fundamental differences in the average position properties of eigenstates versus classical trajectories.

Contribution

It provides a detailed analysis of a non-compact billiard where quantum and classical ergodic properties diverge, highlighting the impact of non-compactness on ergodic behavior.

Findings

Quantum average of position is finite on eigenstates.

Classical time average of position is unbounded.

Quantum and classical ergodicity differ in non-compact billiards.

Abstract

Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-\alpha}$, $\alpha \in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.