Small scale quantum ergodicity in negatively curved manifolds

In this paper, we investigate quantum ergodicity in negatively curved manifolds. We consider the symbols depending on a semiclassical parameter h with support shrinking down to a point as $h\to 0$. The rate of shrinking is a power of $\mid \text{log}h\mid$. This extends the asymptotic equidistribution of quantum ergodic eigenfunctions to a logarithmical scale. (paper)