Abstract
We study the number of bouncing ball modes 
in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically 
for 
, where 
depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to 
, which corresponds to the leading term in Weyl's law for the mean behaviour of the counting function of eigenstates. This result shows that one can come arbitrarily close to violating quantum ergodicity. We compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement.