  This short note proves that a Laplacian cannot be quantum uniquely ergodic if
it possesses a quasimode of order zero which (i) has a singular limit, and (ii)
is a linear combination of a uniformly bounded number of eigenfunctions (modulo
an o(1) error). Bouncing ball quasimodes of the stadium are believed to have
this property (E.J. Heller et al) and so are analogous quasimodes recently
constructed by H. Donnelly on certain non-positively curved surfaces. The main
ingredient is the proof that all sequences of off-diagonal matrix elements of
QUE systems with vanishing spectral gaps tend to zero.
