On the Rate of Quantum Ergodicity in Euclidean Billiards

For a large class of quantized ergodic ows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdi ere and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we rst give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodiierential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We nd that in connguration space the rate of quantum ergodicity is strongly innuenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these eeects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized uctuations of the expectation values around their mean, and nd good agreement with a Gaussian distribution. y