Ergodic properties of infinite quantum harmonic crystals: An analytic approach
- Dipartimento di Matematica, Universita di Bologna, 40127 Bologna (Italy)
- Universite de Paris-Nord, Departement de Mathematiques, CNRS-URA 742, 93430 Villetaneuse (France)
We prove that the quantum dynamics of a class of infinite harmonic crystals becomes ergodic and mixing in the following sense: if {ital H}{sub {ital m}} is the {ital m}-particle Schr{umlt o}dinger operator, {omega}{sub {beta},{ital m}}({ital A})=Tr({ital A}exp{minus}{beta}{ital H}{sub {ital m}})/Tr(exp{minus}{beta}{ital H}{sub {ital m}}) the corresponding quantum Gibbs distribution over the observables {ital A},{ital B},{psi}{sub {ital m},{lambda}} the coherent states in the {ital m}th particle Hilbert space, {ital g}{sub {ital m}},{lambda}=(exp{minus}{beta}{ital H}{sub {ital m}}){psi}{sub {ital m},{lambda}} then lim{sub {ital t}{r_arrow}{infinity}}lim{sub {ital n}{r_arrow}{infinity}}lim{sub {ital m}{r_arrow}{infinity}}(1/{ital T}){integral}{sup {ital T}}{sub 0}{l_angle}{ital e}{sup {ital iH}} {sub {ital n}}{sub {ital t}}{ital Ae}{sup {minus}{ital iH}}{sub {ital n}}{sub {ital t}}{psi}{sub {ital m},{lambda}},{psi}{sub {ital m},{lambda}}{r_angle}{ital dt=lim{sub {ital m}{r_arrow}{infinity}}} {omega}{sub {beta},{ital m}}({ital A}) if the classical infinite dynamics is ergodic, and lim{sub {ital t}{r_arrow}{infinity}}lim{sub {ital n}{r_arrow}{infinity}}lim{sub {ital m}{r_arrow}{infinity}}{omega}{sub {beta},{ital m}}({ital e}{sup /iiH} {sub {ital nt}}{ital Ae} {sup {minus}{ital iH}} {sub {ital nt}}{ital B})=lim{sub {ital m}{r_arrow}{infinity}}{omega}{sub {beta},{ital m}}({ital A})lim{sub {ital m}{r_arrow}{infinity}}{omega}{sub {beta},{ital m}}({ital B}) if it is in addition mixing. The classical ergodicity and mixing properties are recovered as {h_bar}{r_arrow}0, and lim{sub {ital m}{r_arrow}{infinity}}{omega}{sub {beta},{ital m}}({ital A}) turns out to be the average over a classical Gibbs measure of the symbol generating {ital A} under Weyl quantization. {copyright} {ital 1996 American Institute of Physics.}
- OSTI ID:
- 390483
- Journal Information:
- Journal of Mathematical Physics, Vol. 37, Issue 10; Other Information: PBD: Oct 1996
- Country of Publication:
- United States
- Language:
- English
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