Schrodinger group along the horocycles and Egorov's theorem on compact hyperbolic surfaces

Abstract

In this paper, we study Egorov&#39;s theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch&#39;s quantization. In this quantization, the remainder in Egorov&#39;s theorem for eigenfunctions is given explicitly by the second derivatives of the symbols along the weak stable foliation of the geodesic flow. Forward in time, we show that Egorov&#39;s theorem for eigenfunctions is valid to all logarithmic times, far beyond the Ehrenfest time T_E=|log h|/2, where h going to 0 is the semiclassical parameter. Backward in time, we introduce the Schrodinger group along the stable horocycles, which absorbs the exponentially growing part of the remainder. We show that Egorov&#39;s theorem for eigenfunctions is valid up to -T_E+T_h for any Th going to infinity with 0<=T_h<=T_E. Moreover, the limit of the remainder in Egorov&#39;s theorem at the Ehrenfest time -T_E is exactly the difference between the semiclassical measure and the Liouville measure, tested against the symbols propagated by the Schrodinger group. Consequently, Egorov&#39;s theorem for a sequence of eigenfunctions holds at -T_E (and in fact, at -T_E-T for any fixed T in R if and only if the sequence tends to be equidistributed in the phase space. Thus, the quantum unique ergodicity (QUE) conjecture is equivalent to this Egorov property holding for every eigenbasis. Furthermore, QUE holds with a polynomial rate if and only if Egorov&#39;s theorem holds at -T_E with a polynomial remainder. In this case, Egorov&#39;s theorem holds beyond -T_E with a polynomial remainder, and the stronger the rate is in QUE, the longer the time is in Egorov&#39;s theorem.