Affiliation:
1. Department of Mathematics, Northwestern University, Evanston, IL 60208-2370, USA
Abstract
We consider a sequence
of finite-dimensional Hilbert spaces of dimensions
. 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
may be identified with
U
(
d
N
), and a random orthonormal basis of
is a choice of a random sequence
U
N
∈
U
(
d
N
) from the product of normalized Haar measures. We prove that if
and if
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
is the space of spherical harmonics on
S
2
. In particular, it holds on the flat torus
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.
Subject
General Physics and Astronomy,General Engineering,General Mathematics
Reference16 articles.
1. Quantum ergodicity on the sphere
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