Quantum unique ergodicity

TL;DR

This paper demonstrates that certain quasimodes prevent a Laplacian from being quantum uniquely ergodic, especially when these quasimodes have singular limits and are composed of a bounded number of eigenfunctions.

Contribution

It establishes a new criterion showing that specific quasimodes obstruct quantum unique ergodicity in Laplacian systems.

Findings

Off-diagonal matrix elements tend to zero in QUE systems with vanishing spectral gaps.

Bouncing ball quasimodes in stadiums are likely to prevent QUE.

Constructed quasimodes on non-positively curved surfaces also serve as obstructions.

Abstract

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an o(1) error). Bouncing ball quasimodes of the stadium are believed to have this property (E.J. Heller et al) and so are analogous quasimodes recently constructed by H. Donnelly on certain non-positively curved surfaces. The main ingredient is the proof that all sequences of off-diagonal matrix elements of QUE systems with vanishing spectral gaps tend to zero.