Quantum Localization for a Strongly Classically Chaotic System

The classical stadium billiard, which is known to be ergodic and strongly mixing, is shown to have strongly localized quantum eigenstates which persist up to infinite energy, or alternately which survive the $\hslash \to 0$ limit. Consistent with the theorems of Schnirlman, Zelditch, and Colin de Verdiere, the states which we show are highly localized are collectively of zero measure (though there are infinitely many) as $E\to \infty$ or as $\hslash \to 0$. However, more relevant to the experimental world is the fact that such localized states make up a finite and calculable fraction of the quantum eigenstates at finite energy.