Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

Should one expect sequences of eigenfunctions to be
Lagrangian distributions?
NO!! It is very rare.
It is true for joint eigenfunctions of commuting operators such as
∂
∂x j
on R n /Z n ; or ∆ and L 3 = ∂ ∂θ (x 3 -axis rotations) for S 2 .
These are quantum integrable Laplacians: ∆ commutes with n − 1
other independent operators in dimension n. E.g ✷ for
Schwarzchild spacetime.
But these integrable systems are rare. Almost always,
eigenfunctions are not at all like Lagrangian distributions. When
the geodesic flow is ergodic they behave like “random waves”.