# Short examples that are/are not quantum-ergodic

+ 6 like - 0 dislike
202 views

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
retagged Jun 17, 2016
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

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.