Note on quantum unique ergodicity

We prove that (near) off-diagonal matrix elements ⟨ A φ i , φ j ⟩ A φ _i, φ _j ( i ≠ j i = j ) of pseudodifferential operators relative to eigenfunctions of quantum unique- ly ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.