TY - JOUR

T1 - Quantum ergodicity of random orthonormal bases of spaces of high dimension

AU - Zelditch, Steve

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2014/1/28

Y1 - 2014/1/28

N2 - 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.

AB - 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.

KW - Laplace eigenfunctions

KW - Quantum ergodcity

KW - Random orthonormal basis

UR - http://www.scopus.com/inward/record.url?scp=84890445237&partnerID=8YFLogxK

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U2 - 10.1098/rsta.2012.0511

DO - 10.1098/rsta.2012.0511

M3 - Article

C2 - 24344341

AN - SCOPUS:84890445237

VL - 372

JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8428

IS - 2007

M1 - 20120511

ER -