Ergodic properties of infinite quantum harmonic crystals: An analytic approach

We prove that the quantum dynamics of a class of infinite harmonic crystals becomes ergodic and mixing in the following sense: if H-m is the m-particle Schrodinger operator, omega(beta,m) (A)=Tr(A exp-beta H-m)/Tr(exp-beta H-m) the corresponding quantum Gibbs distribution over the observables A, B, psi(m,lambda) the coherent states in he mth particle Hilbert space, g(m,lambda)=(exp=beta H-m) psi(m,lambda) then lim(t-->infinity) lim(n-->infinity) lim(m-->infinity)(1/T) integral(0)(T)[e(iHnt)psi(m,lambda), psi(m,lambda)] dt = lim(m-->infinity) omega(beta,m) (A) if the classical infinite dynamics is ergodic, and lim(t-->infinity) lim(n-->infinity) lim(m-->infinity) omega(beta,m) (e(iHnt)Ae(-iHnt)B) = lim(m-->infinity) omega(beta,m) (A) lim(m-->infinity) omega(beta,m) (B) if it is in addition mixing. The classical ergodicity and mixing properties are recovered as h-->0, and lim(m-->infinity) omega(beta,m) (A) turns out to be the average over a classical Gibbs measure of the symbol generating A under Weyl quantization. (C) 1996 American Institute of Physics.