Short examples that are/are not quantum-ergodic

Question

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.

This post imported from StackExchange MathOverflow at 2016-06-17 12:23 (UTC), posted by SE-user user48339

Comments

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.

This post imported from StackExchange MathOverflow at 2016-06-17 12:23 (UTC), posted by SE-user Asaf

Answer 1

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.

Zelditch showed that if you pick an ONB of eigenfunctions at random, they will have quantum ergodic behavior.

This post imported from StackExchange MathOverflow at 2016-06-17 12:23 (UTC), posted by SE-user forevenone