RATE OF QUANTUM ERGODICITY IN EUCLIDEAN BILLIARDS

For a large class of quantized ergodic flows the quantum ergodicity th eorem states that almost all eigen functions become equidistributed in the semiclassical limit. In this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observa bles in terms of pseudodifferential operators and show that it is equi valent to the semiclassical eigenfunction hypothesis for the Wigner fu nction in the case of ergodic systems. Of great importance is the rate by which the quantum-mechanical expectation values of an observable t end to their mean value. This is studied numerically for three Euclide an billiards (stadium, cosine, and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quant um ergodicity is strongly influenced by localized eigenfunctions such as bouncing-ball modes or scarred eigenfunctions. We give a detailed d iscussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observ e a slower decay. We also study the suitably normalized fluctuations o f the expectation values around their mean and find good agreement wit h a Gaussian distribution.