On the rate of quantum ergodicity I: Upper bounds
On the rate of quantum ergodicity I: Upper bounds
复制标题
关于量子遍历率 I：上限
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
S. Zelditch
中科院分区：
文献类型：
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作者：
S. Zelditch
摘要量子遍历中的一个问题是估计和的衰减率
$$S_k(lambda；A)=frac{1}{{N(Lambda)}}SumLimits_{SQRT{lambda_j}leqq lambda}{Left|{(avarphi_j，varphi_j)-ar sigma_A}
晚上|^k}$$
关于具有遍历测地流的紧致黎曼流形(M，g)。这里，{λj，ϕj}是(M，g)的Δ的光谱数据，A是0阶ψDO，
$$AR SIGMA_A$$
是其主要符号的(刘维尔)平均值
$$N(Lambda)=#{j：sqrt{lambda_j}leqq lambda}$$
。文[S，Z.1，CV.1]证明了Sk(λ；A)=o(1)。本文的目的是证明(可能可变的)负曲率流形上的Sk(λ；A)=O((Logλ)−k/2))。主要的新内容是此类空间([R，Si])上测地线流的中心极限定理。
AbstractOne problem in quantum ergodicity is to estimate the rate of decay of the sums
$$S_k (lambda ;A) = frac{1}{{N(lambda )}}sumlimits_{sqrt {lambda _j } leqq lambda } {left| {(Avarphi _j ,varphi _j ) - ar sigma _A }
ight|^k } $$
on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λj,ϕj} are the spectral data of the Δ of(M, g), A is a 0-th order ψDO,
$$ar sigma _A $$
is the (Liouville) average of its principal symbol and
$$N(lambda ) = # { j:sqrt {lambda _j } leqq lambda } $$
. ThatSk(λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatSk(λ;A)=O((logλ)−k/2) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).