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