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

We consider distributions of diagonal matrix elements for smooth observables in systems whose classical phase space has a mixture of chaotic and nearly integrable regions. The quantum distributions agree very well with distributions obtained from classical trajectory segments whose length is the Heisenberg time. Non-Gaussian wings in the distributions can be linked to classical trapping in certain parts of phase space, sometimes connected to islands, but also to regions separated by other barriers to transport. Thus classical deviations from ergodicity an quantitatively reflected in quantum matrix elements. The relation to scars is discussed. [S1063-651X(99)04205-1].