# Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?

Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost all of the eigenfunctions of its Laplacian operator equidistribute, while it's quantum unique ergodic if absolutely all of them do. A summary can be found here: http://www.austms.org.au/Publ/Gazette/2011/Jul11/TechPaperHassell.pdf.

• Well billiards are in general not QUE - see Hassel - "Ergodic billiards that are not quantum unique ergodic", if I recall correctly the proof is short. – Asaf Apr 19 '16 at 8:35
• A little irrelevant to the question perhaps, but when reading that article I am struck by a number of incorrect statements: a homeomorphism of a compact space $X$ with a periodic orbit cannot be uniquely ergodic, a uniquely ergodic homeomorphism of a compact topological space cannot have a non-dense orbit. A counterexample to both claims is $x \mapsto x+1$ on the one-point compactification of $\mathbb{Z}$. – Ian Morris Jul 16 '16 at 17:27

The unit sphere $S^n$ with standard round metrics is certainly not QUE, due to the fact that we have eigenfuntions like Zonal functions which always concentrates near points and highest weight spherical harmonics which always concentrates near geodesics. I think it should be true but I don't know though if anyone has proved that spheres are not quantum ergodic.