Some remarks on the off‐diagonal asymptotics in quantum ergodicity

Article type: Research Article

Authors: Tate, Tatsuya^;

Affiliations: Department of Mathematics, Tohoku University, Aoba‐ku, Sendai, 980‐77, Japan

Note: [] The author is a Research Fellow of the Japan Society for the Promotion of Science. Present address: Department of Mathematics, Faculty of Science and Technology, Keio University, 3‐14‐1, Hiyoshi, Kohoku‐ku, Yokohama, 223‐8522, Japan. E‐mail: [email protected].

Abstract: 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 {\rm d}m_{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.

Journal: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 289-296, 1999