Quantum ergodicity of random orthonormal bases of spaces of high dimension

We consider a sequence HN of finite-dimensional Hilbert spaces of dimensions dN→∞. Motivating examples are eigenspaces, or spaces of quasimodes, 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 N HN is a choice of a random sequence UN ε 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 S2. 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.