Department of Mathematics, Northwestern University, , Evanston, IL 60208-2370, USA.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20120511. doi: 10.1098/rsta.2012.0511. Print 2014 Jan 28.
We consider a sequence HN of finite-dimensional Hilbert spaces of dimensions dN → ∞. Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of HN may be identified with U(dN), and a random orthonormal basis of is a choice of a random sequence UN∈U(dN) from the product of normalized Haar measures. We prove that if dN → ∞ and if(1/dN)TrA|HN tends to a unique limit state ω(A), then almost surely an orthonormal basis is quantum ergodic with limit state ω(A). This generalizes an earlier result of the author in the case where HN is the space of spherical harmonics on S(2). In particular, it holds on the flat torus Rd/Zd if d≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.
我们考虑一个有限维希尔伯特空间序列 HN，维度 dN→∞。激励性的例子是紧黎曼流形上拉普拉斯或薛定谔算子的特征空间或拟模空间。HN 的厄米正交基集可以与 U(dN) 相媲美，而的随机正交基是从归一化 Haar 测度的积中选择的随机序列 UN∈U(dN)。我们证明，如果 dN→∞，并且(1/dN)TrA|HN 趋于唯一的极限态 ω(A)，则几乎必然地正交基是具有极限态 ω(A)的量子遍历的。这推广了作者在 HN 是 S(2)上的球谐空间的情况下的早期结果。特别地，如果 d≥5，则在平坦的环面 Rd/Zd 上成立，并且表明在相对较小的维度中，可以从量子遍历的基中合成高度局域的正交基，反之亦然。
