ON THE RATE OF QUANTUM ERGODICITY ON HYPERBOLIC SURFACES AND FOR BILLIARDS

The rate of quantum ergodicity is studied for three strongly chaotic ( Anosov) systems. The quantal eigenfunctions on a compact Riemannian su rface of genus g = 2 and of two triangular billiards on a surface of c onstant negative curvature are investigated. One of the triangular bil liards belongs to the class of arithmetic systems. There are no peculi arities observed in the arithmetic system concerning the rate of quant um ergodicity. This contrasts to the peculiar behaviour with respect t o the statistical properties of the quantal levels. It is demonstrated that the rate of quantum ergodicity in the three considered systems f its well with the known upper and lower bounds. Furthermore, Sarnak's conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems. Copyright (C) 1998 Elsev ier Science B.V.