Some remarks on the off-diagonal asymptotics in quantum ergodicity

Our purpose of this paper is to estimate the rate of the decay of the off-diagonal asymptotics in Sunada (preprint) in case where the Hamilton flow is of Anosov type. For the sake of this, we use the central limit theorem for transitive Anosov flows (Sinai, Soviet Math. Dokl. 1 (1960), 983-987; Ratner, Israel J. Math. 16 (1973), 181-197; Zelditch, Comm. Math. Phys. 160 (1994), 81-92). Also it is shown that if the Hamilton flow has homogeneous Lebesgue spectrum, then the measure dm_A associated with a pseudodifferential operator A, which is introduced by Zelditch (J. Funct. Anal. 140 (1996), 68-86), is absolutely continuous with respect to Lebesgue measure.