Journal of Statistical Physics, Vol. 116, Nos. 1/4, August 2004, p. 821-830.
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, a from (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.