Phase space localization and matrix element distributions in systems with mixed classical phase space

We consider distributions of diagonal matrix elements for smooth observable s in systems whose classical phase space has a mixture of chaotic and nearl y integrable regions. The quantum distributions agree very well with distri butions obtained from classical trajectory segments whose length is the Hei senberg time. Non-Gaussian wings in the distributions can be linked to clas sical trapping in certain parts of phase space, sometimes connected to isla nds, but also to regions separated by other barriers to transport. Thus cla ssical deviations from ergodicity an quantitatively reflected in quantum ma trix elements. The relation to scars is discussed. [S1063-651X(99)04205-1].