Localization in infinite billiards: A comparison between quantum and classical ergodicity

Consider the non-compact billiard in the first quandrant bounded by the positive x-semiaxis, the positive y-semiaxis and the graph of f(x) = (x + 1)(-a), alpha is an element of (1, 2). Although the Schnirelman Theorem holds, the quantum average of the position x is finite on any eigenstate, while classical ergodicity entails that the classical time average of x is unbounded.