2022-08-09T04:20:18Zhttps://soar-ir.repo.nii.ac.jp/oaioai:soar-ir.repo.nii.ac.jp:000182552021-09-02T06:23:03Z1169:1170Measure theoretical approach to recurrent properties for quantum dynamicsOtobe, YoshikiSasaki, ItaruPoincare's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincare's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrodinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.ArticleJOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 44(46):465209 (2011)journal articleIOP PUBLISHING LTD2011-11-18application/pdfJOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL46444652091751-8113AA12185372https://soar-ir.repo.nii.ac.jp/record/18255/files/Measure_theoretical_approach_recurrent_properties.pdfeng10.1088/1751-8113/44/46/465209https://doi.org/10.1088/1751-8113/44/46/465209This is an author-created, un-copyedited version of an article accepted for publication in JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version is available online at 10.1088/1751-8113/44/46/465209. © 2011 IOP Publishing Ltd