Quantum Leaks

TL;DR

This paper demonstrates that eigenfunctions on specific non-compact finite-area domains can localize at infinity, challenging quantum unique ergodicity, through elementary constructions of bouncing ball quasimodes with small discrepancy.

Contribution

It introduces a simple method to construct bouncing ball quasimodes that show eigenfunction localization, providing counterexamples to quantum unique ergodicity in certain systems.

Findings

Eigenfunctions can localize at infinity on non-compact domains.

Bouncing ball quasimodes have discrepancy smaller than mean level spacing.

Quantum unique ergodicity does not hold universally for these systems.

Abstract

We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.