Localization of eigenfunctions in the stadium billiard

We present a systematic survey of scarring and symmetry effects in the stadium billiard. The localization of individual eigenfunctions in Husimi phase space is studied first, and it is demonstrated that on average there is more localization than can be accounted for by random-matrix theory, even after removal of bouncing-ball stares and visible scars. A major point of the paper is that symmetry considerations, including parity and time-reversal symmetries, enter to influence the total amount of localization. The properties of the local density of states are also investigated, as a function of phase space location. Aside from the bouncing-ball region of phase space, excess localization is found on short periodic orbits and along certain symmetry-related lines: the origin of all these sources of localization is discussed quantitatively and comparison is made with analytical predictions. Scarring is observed to be present in all the energy ranges considered. In light of our results, the excess localization in individual eigenstates is interpreted as being primarily due to symmetry effects.